INTERNATIONAL GONGRESS 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z">R-algebra such that
(i) A R K K [ X 1 , 1 X 1 , , X n , 1 X n ] A ⊗ R K ≅ K X 1 , 1 X 1 , … , X n , 1 X n Aox_(R)K~=K[X_(1),(1)/(X_(1)),dots,X_(n),(1)/(X_(n))]A \otimes_{R} K \cong K\left[X_{1}, \frac{1}{X_{1}}, \ldots, X_{n}, \frac{1}{X_{n}}\right]A⊗RK≅K[X1,1X1,…,Xn,1Xn],
(ii) for each height-one prime ideal P P PPP of R , A R k ( P ) k ( P ) [ X 1 , 1 X 1 , R , A ⊗ R k ( P ) ≅ k ( P ) X 1 , 1 X 1 , … R,Aox_(R)k(P)~=k(P)[X_(1),(1)/(X_(1)),dots:}R, A \otimes_{R} k(P) \cong k(P)\left[X_{1}, \frac{1}{X_{1}}, \ldots\right.R,A⊗Rk(P)≅k(P)[X1,1X1,…, X n , 1 X n ] X n , 1 X n {:X_(n),(1)/(X_(n))]\left.X_{n}, \frac{1}{X_{n}}\right]Xn,1Xn]
Then A A AAA is a locally Laurent polynomial algebra in n n nnn variables over R R RRR, i.e.,
A m = R m [ X 1 , 1 X 1 , , X n , 1 X n ] A m = R m X 1 , 1 X 1 , … , X n , 1 X n A_(m)=R_(m)[X_(1),(1)/(X_(1)),dots,X_(n),(1)/(X_(n))]A_{m}=R_{m}\left[X_{1}, \frac{1}{X_{1}}, \ldots, X_{n}, \frac{1}{X_{n}}\right]Am=Rm[X1,1X1,…,Xn,1Xn]
and is of the form B [ I 1 ] B I − 1 B[I^(-1)]B\left[I^{-1}\right]B[I−1], where B B BBB is the symmetric algebra of a projective R R RRR-module Q Q QQQ of rank n , Q n , Q n,Qn, Qn,Q is a direct sum of finitely generated projective R R RRR-modules of rank one, and I I III is an invertible ideal of B B BBB.

5. EPIMORPHISM PROBLEM

The Epimorphism Problem for hypersurfaces asks the following fundamental question:
Question 4. Let k k kkk be a field and f B = k [ n ] f ∈ B = k [ n ] f in B=k^([n])f \in B=k^{[n]}f∈B=k[n] for some integer n 2 n ≥ 2 n >= 2n \geq 2n≥2. Suppose
B / ( f ) k [ n 1 ] B / ( f ) ≅ k [ n − 1 ] B//(f)~=k^([n-1])B /(f) \cong k^{[n-1]}B/(f)≅k[n−1]
Does this imply that B = k [ f ] [ n 1 ] B = k [ f ] [ n − 1 ] B=k[f]^([n-1])B=k[f]^{[n-1]}B=k[f][n−1], i.e., is f f fff a coordinate in B B BBB ?
This problem is generally known as the Epimorphism Problem. It is an open problem and is regarded as one of the most challenging and celebrated problems in the area of affine algebraic geometry (see [ 38 , 69 , 75 , 77 ] [ 38 , 69 , 75 , 77 ] [38,69,75,77][38,69,75,77][38,69,75,77] for useful surveys).
The first major breakthrough on Question 4 was achieved during 1974-1975, independently, by Abhyankar-Moh and Suzuki [5,86]. They showed that Question 4 has an affirmative answer when k k kkk is a field of characteristic zero and n = 2 n = 2 n=2n=2n=2. Over a field of positive characteristic, explicit examples of nonrectifiable epimorphisms from k [ X , Y ] k [ X , Y ] k[X,Y]k[X, Y]k[X,Y] to k [ T ] k [ T ] k[T]k[T]k[T] (referred to in Section 2) and hence explicit examples of nontrivial lines had already been demonstrated by Segre [83] in 1957 and Nagata [71] in 1971. However, over a field of characteristic zero, we have the following conjecture:
Abhyankar-Sathaye Conjecture. Let k k kkk be a field of characteristic zero and f B = k [ n ] f ∈ B = k [ n ] f in B=k^([n])f \in B=k^{[n]}f∈B=k[n] for some integer n 2 n ≥ 2 n >= 2n \geq 2n≥2. Suppose that B / ( f ) k [ n 1 ] B / ( f ) ≅ k [ n − 1 ] B//(f)~=k^([n-1])B /(f) \cong k^{[n-1]}B/(f)≅k[n−1]. Then B = k [ f ] [ n 1 ] B = k [ f ] [ n − 1 ] B=k[f]^([n-1])B=k[f]^{[n-1]}B=k[f][n−1].
In case n = 3 n = 3 n=3n=3n=3, some special cases have been solved by Sathaye, Russell, and Wright [73,76,79,89]. In [79], Sathaye proved the conjecture for the linear planes, i.e., polynomials F F FFF of the form a Z b a Z − b aZ-ba Z-baZ−b, where a , b k [ X , Y ] a , b ∈ k [ X , Y ] a,b in k[X,Y]a, b \in k[X, Y]a,b∈k[X,Y]. This was further extended by Russell over fields of any characteristic. They proved that
Theorem 5.1. Let F k [ X , Y , Z ] F ∈ k [ X , Y , Z ] F in k[X,Y,Z]F \in k[X, Y, Z]F∈k[X,Y,Z] be such that F = a Z b F = a Z − b F=aZ-bF=a Z-bF=aZ−b, where a ( 0 ) , b k [ X , Y ] a ( ≠ 0 ) , b ∈ k [ X , Y ] a(!=0),b in k[X,Y]a(\neq 0), b \in k[X, Y]a(≠0),b∈k[X,Y], and k [ X , Y , Z ] / ( F ) = k [ 2 ] k [ X , Y , Z ] / ( F ) = k [ 2 ] k[X,Y,Z]//(F)=k^([2])k[X, Y, Z] /(F)=k^{[2]}k[X,Y,Z]/(F)=k[2]. Then there exist X 0 , Y 0 k [ X , Y ] X 0 , Y 0 ∈ k [ X , Y ] X_(0),Y_(0)in k[X,Y]X_{0}, Y_{0} \in k[X, Y]X0,Y0∈k[X,Y] such that k [ X , Y ] = k [ X 0 , Y 0 ] k [ X , Y ] = k X 0 , Y 0 k[X,Y]=k[X_(0),Y_(0)]k[X, Y]=k\left[X_{0}, Y_{0}\right]k[X,Y]=k[X0,Y0] with a k [ X 0 ] a ∈ k X 0 a in k[X_(0)]a \in k\left[X_{0}\right]a∈k[X0] and k [ X , Y , Z ] = k [ X 0 , F ] [ 1 ] k [ X , Y , Z ] = k X 0 , F [ 1 ] k[X,Y,Z]=k[X_(0),F]^([1])k[X, Y, Z]=k\left[X_{0}, F\right]^{[1]}k[X,Y,Z]=k[X0,F][1].
When k k kkk is an algebraically closed field of characteristic p 0 p ≥ 0 p >= 0p \geq 0p≥0, Wright [89] proved the conjecture for polynomials F F FFF of the form a Z m b a Z m − b aZ^(m)-ba Z^{m}-baZm−b with a , b k [ X , Y ] , m 2 a , b ∈ k [ X , Y ] , m ≥ 2 a,b in k[X,Y],m >= 2a, b \in k[X, Y], m \geq 2a,b∈k[X,Y],m≥2 and p m p ∤ m p∤mp \nmid mp∤m. Das and Dutta showed [28, THEOREM 4.5] that Wright's result extends to any field k k kkk. They proved that
Theorem 5.2. Let k k kkk be any field with ch k = p ( 0 ) k = p ( ≥ 0 ) k=p( >= 0)k=p(\geq 0)k=p(≥0) and F = a Z m b k [ X , Y , Z ] F = a Z m − b ∈ k [ X , Y , Z ] F=aZ^(m)-b in k[X,Y,Z]F=a Z^{m}-b \in k[X, Y, Z]F=aZm−b∈k[X,Y,Z] be such that a ( 0 ) , b k [ X , Y ] , m 2 a ( ≠ 0 ) , b ∈ k [ X , Y ] , m ≥ 2 a(!=0),b in k[X,Y],m >= 2a(\neq 0), b \in k[X, Y], m \geq 2a(≠0),b∈k[X,Y],m≥2 and p m p ∤ m p∤mp \nmid mp∤m. Suppose that k [ X , Y , Z ] / ( F ) = k [ 2 ] k [ X , Y , Z ] / ( F ) = k [ 2 ] k[X,Y,Z]//(F)=k^([2])k[X, Y, Z] /(F)=k^{[2]}k[X,Y,Z]/(F)=k[2]. Then there exists X 0 k [ X , Y ] X 0 ∈ k [ X , Y ] X_(0)in k[X,Y]X_{0} \in k[X, Y]X0∈k[X,Y] such that k [ X , Y ] = k [ X 0 , b ] k [ X , Y ] = k X 0 , b k[X,Y]=k[X_(0),b]k[X, Y]=k\left[X_{0}, b\right]k[X,Y]=k[X0,b] with a k [ X 0 ] a ∈ k X 0 a in k[X_(0)]a \in k\left[X_{0}\right]a∈k[X0] and k [ X , Y , Z ] = k [ X , Y , Z ] = k[X,Y,Z]=k[X, Y, Z]=k[X,Y,Z]= k [ F , Z , X 0 ] k F , Z , X 0 k[F,Z,X_(0)]k\left[F, Z, X_{0}\right]k[F,Z,X0].
The condition that p m p ∤ m p∤mp \nmid mp∤m is necessary in Theorem 5.2 (cf. [28, REMARK 4.6]).
Most of the above cases are covered by the following generalization due to Russell and Sathaye [76, THEOREM 3.6]:
Theorem 5.3. Let k k kkk be a field of characteristic zero and let
F = a m Z m + a m 1 Z m 1 + + a 1 Z + a 0 k [ X , Y , Z ] F = a m Z m + a m − 1 Z m − 1 + ⋯ + a 1 Z + a 0 ∈ k [ X , Y , Z ] F=a_(m)Z^(m)+a_(m-1)Z^(m-1)+cdots+a_(1)Z+a_(0)in k[X,Y,Z]F=a_{m} Z^{m}+a_{m-1} Z^{m-1}+\cdots+a_{1} Z+a_{0} \in k[X, Y, Z]F=amZm+am−1Zm−1+⋯+a1Z+a0∈k[X,Y,Z]
where a 0 , , a m k [ X , Y ] a 0 , … , a m ∈ k [ X , Y ] a_(0),dots,a_(m)in k[X,Y]a_{0}, \ldots, a_{m} \in k[X, Y]a0,…,am∈k[X,Y] are such that GCD ( a 1 , , a m ) k GCD ⁡ a 1 , … , a m ∉ k GCD(a_(1),dots,a_(m))!in k\operatorname{GCD}\left(a_{1}, \ldots, a_{m}\right) \notin kGCD⁡(a1,…,am)∉k. Suppose that
k [ X , Y , Z ] / ( F ) = k [ 2 ] k [ X , Y , Z ] / ( F ) = k [ 2 ] k[X,Y,Z]//(F)=k^([2])k[X, Y, Z] /(F)=k^{[2]}k[X,Y,Z]/(F)=k[2]
Then there exists X 0 k [ X , Y ] X 0 ∈ k [ X , Y ] X_(0)in k[X,Y]X_{0} \in k[X, Y]X0∈k[X,Y] such that k [ X , Y ] = k [ X 0 , b ] k [ X , Y ] = k X 0 , b k[X,Y]=k[X_(0),b]k[X, Y]=k\left[X_{0}, b\right]k[X,Y]=k[X0,b] with a m k [ X 0 ] a m ∈ k X 0 a_(m)in k[X_(0)]a_{m} \in k\left[X_{0}\right]am∈k[X0]. Further, k [ X , Y , Z ] = k [ F ] [ 2 ] k [ X , Y , Z ] = k [ F ] [ 2 ] k[X,Y,Z]=k[F]^([2])k[X, Y, Z]=k[F]^{[2]}k[X,Y,Z]=k[F][2].
Thus, for k [ X , Y , Z ] k [ X , Y , Z ] k[X,Y,Z]k[X, Y, Z]k[X,Y,Z], the Abhyankar-Sathaye conjecture remains open for the case when GCD ( a 1 , , a m ) = 1 GCD ⁡ a 1 , … , a m = 1 GCD(a_(1),dots,a_(m))=1\operatorname{GCD}\left(a_{1}, \ldots, a_{m}\right)=1GCD⁡(a1,…,am)=1.
A common theme in most of the partial results proved in the Abhyankar-Sathaye conjecture for k [ X , Y , Z ] k [ X , Y , Z ] k[X,Y,Z]k[X, Y, Z]k[X,Y,Z] is that, if F F FFF is considered as a polynomial in Z Z ZZZ, then the coordinates of k [ X , Y ] k [ X , Y ] k[X,Y]k[X, Y]k[X,Y] can be so chosen that the coefficient of Z Z ZZZ becomes a polynomial in X X XXX. The Abhyankar-Sathaye conjecture for k [ X , Y , Z ] k [ X , Y , Z ] k[X,Y,Z]k[X, Y, Z]k[X,Y,Z] can now be split into two parts.
Question 4A. Let k k kkk be a field of characteristic zero and let
F = a m Z m + a m 1 Z m 1 + + a 1 Z + a 0 k [ X , Y , Z ] F = a m Z m + a m − 1 Z m − 1 + ⋯ + a 1 Z + a 0 ∈ k [ X , Y , Z ] F=a_(m)Z^(m)+a_(m-1)Z^(m-1)+cdots+a_(1)Z+a_(0)in k[X,Y,Z]F=a_{m} Z^{m}+a_{m-1} Z^{m-1}+\cdots+a_{1} Z+a_{0} \in k[X, Y, Z]F=amZm+am−1Zm−1+⋯+a1Z+a0∈k[X,Y,Z]
where a 0 , , a m k [ X , Y ] a 0 , … , a m ∈ k [ X , Y ] a_(0),dots,a_(m)in k[X,Y]a_{0}, \ldots, a_{m} \in k[X, Y]a0,…,am∈k[X,Y]. Suppose that k [ X , Y , Z ] / ( F ) = k [ 2 ] k [ X , Y , Z ] / ( F ) = k [ 2 ] k[X,Y,Z]//(F)=k^([2])k[X, Y, Z] /(F)=k^{[2]}k[X,Y,Z]/(F)=k[2]. Does there exist X 0 k [ X , Y ] X 0 ∈ k [ X , Y ] X_(0)in k[X,Y]X_{0} \in k[X, Y]X0∈k[X,Y] such that k [ X , Y ] = k [ X 0 ] [ 1 ] k [ X , Y ] = k X 0 [ 1 ] k[X,Y]=k[X_(0)]^([1])k[X, Y]=k\left[X_{0}\right]^{[1]}k[X,Y]=k[X0][1] with a m k [ X 0 ] a m ∈ k X 0 a_(m)in k[X_(0)]a_{m} \in k\left[X_{0}\right]am∈k[X0] ?
Question 4B. Let k k kkk be a field of characteristic zero and suppose
F = a m ( X ) Z m + a m 1 Z m 1 + + a 1 Z + a 0 k [ X , Y , Z ] F = a m ( X ) Z m + a m − 1 Z m − 1 + ⋯ + a 1 Z + a 0 ∈ k [ X , Y , Z ] F=a_(m)(X)Z^(m)+a_(m-1)Z^(m-1)+cdots+a_(1)Z+a_(0)in k[X,Y,Z]F=a_{m}(X) Z^{m}+a_{m-1} Z^{m-1}+\cdots+a_{1} Z+a_{0} \in k[X, Y, Z]F=am(X)Zm+am−1Zm−1+⋯+a1Z+a0∈k[X,Y,Z]
where a 0 , , a m 1 k [ X , Y ] a 0 , … , a m − 1 ∈ k [ X , Y ] a_(0),dots,a_(m-1)in k[X,Y]a_{0}, \ldots, a_{m-1} \in k[X, Y]a0,…,am−1∈k[X,Y] and a m k [ X ] a m ∈ k [ X ] a_(m)in k[X]a_{m} \in k[X]am∈k[X]. Suppose that k [ X , Y , Z ] / ( F ) = k [ 2 ] k [ X , Y , Z ] / ( F ) = k [ 2 ] k[X,Y,Z]//(F)=k^([2])k[X, Y, Z] /(F)=k^{[2]}k[X,Y,Z]/(F)=k[2]. Does this imply that k [ X , Y , Z ] = k [ F ] [ 2 ] k [ X , Y , Z ] = k [ F ] [ 2 ] k[X,Y,Z]=k[F]^([2])k[X, Y, Z]=k[F]^{[2]}k[X,Y,Z]=k[F][2] ?
Sangines Garcia in his PhD thesis [78] answered Question 4A affirmatively for the case m = 2 m = 2 m=2m=2m=2. In [21], Bhatwadekar and the author have given an alternative proof of this result of Garcia.
When k k kkk is any field, as a partial generalization of Theorem 5.1 and Question 4B in four variables, the author proved the Abhyankar-Sathaye conjecture for a polynomial F F FFF of the form X m Y F ( X , Z , T ) k [ X , Y , Z , T ] X m Y − F ( X , Z , T ) ∈ k [ X , Y , Z , T ] X^(m)Y-F(X,Z,T)in k[X,Y,Z,T]X^{m} Y-F(X, Z, T) \in k[X, Y, Z, T]XmY−F(X,Z,T)∈k[X,Y,Z,T]. This was one of the consequences of her general investigation on the ZCP [46]. In the process, she related it with other central problems on affine spaces like the affine fibration problem and the ZCP. The author has proved equivalence of ten statements, some of which involve an invariant introduced by Derksen, which is called the Derksen invariant.
The Derksen invariant of an integral domain B B BBB, denoted by D K ( B ) D K ( B ) DK(B)\mathrm{DK}(B)DK(B), is defined as the smallest subring of B B BBB generated by the kernel of D D DDD, where D D DDD varies over the set of all locally nilpotent derivations of B B BBB.
Theorem 5.4. Let k k kkk be a field of any characteristic and A A AAA an integral domain defined by
A = k [ X , Y , Z , T ] / ( X m Y F ( X , Z , T ) ) , where m > 1 A = k [ X , Y , Z , T ] / X m Y − F ( X , Z , T ) ,  where  m > 1 A=k[X,Y,Z,T]//(X^(m)Y-F(X,Z,T)),quad" where "m > 1A=k[X, Y, Z, T] /\left(X^{m} Y-F(X, Z, T)\right), \quad \text { where } m>1A=k[X,Y,Z,T]/(XmY−F(X,Z,T)), where m>1
Let x , y , z x , y , z x,y,zx, y, zx,y,z, and t t ttt denote, respectively, the images of X , Y , Z X , Y , Z X,Y,ZX, Y, ZX,Y,Z, and T T TTT in A A AAA. Set f ( Z , T ) := f ( Z , T ) := f(Z,T):=f(Z, T):=f(Z,T):= F ( 0 , Z , T ) F ( 0 , Z , T ) F(0,Z,T)F(0, Z, T)F(0,Z,T) and G := X m Y F ( X , Z , T ) G := X m Y − F ( X , Z , T ) G:=X^(m)Y-F(X,Z,T)G:=X^{m} Y-F(X, Z, T)G:=XmY−F(X,Z,T). Then the following statements are equivalent:
(i) k [ X , Y , Z , T ] = k [ X , G ] [ 2 ] k [ X , Y , Z , T ] = k [ X , G ] [ 2 ] k[X,Y,Z,T]=k[X,G]^([2])k[X, Y, Z, T]=k[X, G]^{[2]}k[X,Y,Z,T]=k[X,G][2].
(ii) k [ X , Y , Z , T ] = k [ G ] [ 3 ] k [ X , Y , Z , T ] = k [ G ] [ 3 ] k[X,Y,Z,T]=k[G]^([3])k[X, Y, Z, T]=k[G]^{[3]}k[X,Y,Z,T]=k[G][3].
(iii) A = k [ x ] [ 2 ] A = k [ x ] [ 2 ] A=k[x]^([2])A=k[x]^{[2]}A=k[x][2].
(iv) A = k [ 3 ] A = k [ 3 ] A=k^([3])A=k^{[3]}A=k[3].
(v) A [ ] k k [ + 3 ] A [ ℓ ] ≅ k k [ ℓ + 3 ] quadA^([ℓ])~=_(k)k^([ℓ+3])\quad A^{[\ell]} \cong_{k} k^{[\ell+3]}A[ℓ]≅kk[ℓ+3] for some integer 0 ℓ ≥ 0 ℓ >= 0\ell \geq 0ℓ≥0 and D K ( A ) k [ x , z , t ] D K ( A ) ≠ k [ x , z , t ] DK(A)!=k[x,z,t]\mathrm{DK}(A) \neq k[x, z, t]DK(A)≠k[x,z,t].
(vi) A A quad A\quad AA is an A 2 A 2 A^(2)\mathbb{A}^{2}A2-fibration over k [ x ] k [ x ] k[x]k[x]k[x] and D K ( A ) k [ x , z , t ] D K ( A ) ≠ k [ x , z , t ] DK(A)!=k[x,z,t]\mathrm{DK}(A) \neq k[x, z, t]DK(A)≠k[x,z,t].
(vii) A A AAA is geometrically factorial over k , DK ( A ) k [ x , z , t ] k , DK ⁡ ( A ) ≠ k [ x , z , t ] k,DK(A)!=k[x,z,t]k, \operatorname{DK}(A) \neq k[x, z, t]k,DK⁡(A)≠k[x,z,t] and the canonical map k K 1 ( A ) k ∗ → K 1 ( A ) k^(**)rarrK_(1)(A)k^{*} \rightarrow K_{1}(A)k∗→K1(A) (induced by the inclusion k A k ↪ A k↪Ak \hookrightarrow Ak↪A ) is an isomorphism.
(viii) A is geometrically factorial over k , DK ( A ) k [ x , z , t ] k , DK ⁡ ( A ) ≠ k [ x , z , t ] k,DK(A)!=k[x,z,t]k, \operatorname{DK}(A) \neq k[x, z, t]k,DK⁡(A)≠k[x,z,t] and ( A / x A ) = k ( A / x A ) ∗ = k ∗ (A//xA)^(**)=k^(**)(A / x A)^{*}=k^{*}(A/xA)∗=k∗
(ix) k [ Z , T ] = k [ f ] [ 1 ] k [ Z , T ] = k [ f ] [ 1 ] k[Z,T]=k[f]^([1])k[Z, T]=k[f]^{[1]}k[Z,T]=k[f][1].
(x) k [ Z , T ] / ( f ) = k [ 1 ] k [ Z , T ] / ( f ) = k [ 1 ] k[Z,T]//(f)=k^([1])k[Z, T] /(f)=k^{[1]}k[Z,T]/(f)=k[1] and DK ( A ) k [ x , z , t ] DK ⁡ ( A ) ≠ k [ x , z , t ] DK(A)!=k[x,z,t]\operatorname{DK}(A) \neq k[x, z, t]DK⁡(A)≠k[x,z,t].
The equivalence of (ii) and (iv) provides an answer to Question 4 for the special case of the polynomial X m Y F ( X , Z , T ) X m Y − F ( X , Z , T ) X^(m)Y-F(X,Z,T)X^{m} Y-F(X, Z, T)XmY−F(X,Z,T). The equivalence of (i) and (iii) provides an answer
to a special case of Question 4 4 ′ 4^(')4^{\prime}4′ (stated below) for the ring R = k [ x ] R = k [ x ] R=k[x]R=k[x]R=k[x]. The equivalence of (iii) and (vi) answers Question 3 in a special situation. For more discussions, see [48].
In a remarkable paper Kaliman proved the following result over the field of complex numbers [56]. Later, Daigle and Kaliman extended it over any field k k kkk of characteristic zero [25].
Theorem 5.5. Let k k kkk be a field of characteristic zero. Let F k [ X , Y , Z ] F ∈ k [ X , Y , Z ] F in k[X,Y,Z]F \in k[X, Y, Z]F∈k[X,Y,Z] be such that k [ X , Y , Z ] / ( F λ ) = k [ 2 ] k [ X , Y , Z ] / ( F − λ ) = k [ 2 ] k[X,Y,Z]//(F-lambda)=k^([2])k[X, Y, Z] /(F-\lambda)=k^{[2]}k[X,Y,Z]/(F−λ)=k[2] for almost every λ k λ ∈ k lambda in k\lambda \in kλ∈k. Then k [ X , Y , Z ] = k [ F ] [ 2 ] k [ X , Y , Z ] = k [ F ] [ 2 ] k[X,Y,Z]=k[F]^([2])k[X, Y, Z]=k[F]^{[2]}k[X,Y,Z]=k[F][2].
A general version of Question 4 can be asked as:
Question 4 4 ′ 4^(')4^{\prime}4′. Let R R RRR be a ring and f A = R [ n ] f ∈ A = R [ n ] f in A=R^([n])f \in A=R^{[n]}f∈A=R[n] for some integer n 2 n ≥ 2 n >= 2n \geq 2n≥2. Suppose
A / ( f ) R [ n 1 ] A / ( f ) ≅ R [ n − 1 ] A//(f)~=R^([n-1])A /(f) \cong R^{[n-1]}A/(f)≅R[n−1]
Does this imply that A = R [ f ] [ n 1 ] A = R [ f ] [ n − 1 ] A=R[f]^([n-1])A=R[f]^{[n-1]}A=R[f][n−1], i.e., is f f fff a coordinate in A A AAA ?
There have been affirmative answers to Question 4 4 ′ 4^(')4^{\prime}4′ in special cases by Bhatwadekar, Dutta, and Das [11,13, 28]. Bhatwadekar and Dutta had considered linear planes, i.e., polynomials F F FFF of the form a Z b a Z − b aZ-ba Z-baZ−b, where a , b R [ X , Y ] a , b ∈ R [ X , Y ] a,b in R[X,Y]a, b \in R[X, Y]a,b∈R[X,Y] over a discrete valuation ring R R RRR and proved that special cases of the linear planes are actually variables. Bhatwadekar-Dutta have also shown [12] that a negative answer to Question 4 4 ′ 4^(')4^{\prime}4′ in the case when n = 3 n = 3 n=3n=3n=3 and R R RRR is a discrete valuation ring containing Q Q Q\mathbb{Q}Q will give a negative answer to the affine fibration problem (Question 3 (i)) for the case n = 2 n = 2 n=2n=2n=2 and d = 2 d = 2 d=2d=2d=2. An example of a case of linear planes which remains unsolved is discussed in Section 7.

6. A n A n A^(n)\mathbb{A}^{\boldsymbol{n}}An-FORMS

Let A A AAA be an algebra over a field k k kkk. We say that A A AAA is an A n A n A^(n)\mathbb{A}^{n}An-form over k k kkk if A k L = L [ n ] A ⊗ k L = L [ n ] Aox_(k)L=L^([n])A \otimes_{k} L=L^{[n]}A⊗kL=L[n] for some finite algebraic extension L L LLL of k k kkk. Let A A AAA be an A n A n A^(n)\mathbb{A}^{n}An-form over a field k k kkk.
When n = 1 n = 1 n=1n=1n=1, it is well known that if L | k L k L|_(k)\left.L\right|_{k}L|k is a separable extension, then A = k [ 1 ] A = k [ 1 ] A=k^([1])A=k^{[1]}A=k[1] (i.e., trivial) and that if L | k L k L|_(k)\left.L\right|_{k}L|k is purely inseparable then A A AAA need not be k [ 1 ] k [ 1 ] k^([1])k^{[1]}k[1]. An extensive study of such purely inseparable algebras was made by Asanuma in [8]. Over any field of positive characteristic, the nontrivial purely inseparable A 1 A 1 A^(1)\mathbb{A}^{1}A1-forms can be used to give examples of nontrivial A n A n A^(n)\mathbb{A}^{n}An-forms for any integer n > 1 n > 1 n > 1n>1n>1.
When n = 2 n = 2 n=2n=2n=2 and L | k L k L|_(k)\left.L\right|_{k}L|k is a separable extension, then Kambayashi established that A = k [ 2 ] A = k [ 2 ] A=k^([2])A=k^{[2]}A=k[2] [57]. However, the problem of existence of nontrivial separable A 3 A 3 A^(3)\mathbb{A}^{3}A3-forms is open in general. A few recent partial results on the triviality of separable A 3 A 3 A^(3)\mathbb{A}^{3}A3-forms are mentioned below.
Let A A AAA be an A 3 A 3 A^(3)\mathbb{A}^{3}A3-form over a field k k kkk of characteristic zero and k ¯ k ¯ bar(k)\bar{k}k¯ be an algebraic closure of k k kkk. Then A = k [ 3 ] A = k [ 3 ] A=k^([3])A=k^{[3]}A=k[3] if it satisfies any one of the following:
(1) A A AAA admits a fixed point free locally nilpotent derivation D D DDD (Daigle and Kaliman [25, COROLLARY 3.3]).
(2) A A AAA contains an element f f fff which is a coordinate of A k k ¯ A ⊗ k k ¯ Aox_(k) bar(k)A \otimes_{k} \bar{k}A⊗kk¯ (Daigle and Kaliman [25, PROPOSITION 4.9]).
(3) A A AAA admits an effective action of a reductive algebraic k k kkk-group of positive dimension (Koras and Russell [61, THEOREM c]).
(4) A A AAA admits either a fixed point free locally nilpotent derivation or a nonconfluent action of a unipotent group of dimension two (Gurjar, Masuda, and Miyanishi [51]).
(5) A A AAA admits a locally nilpotent derivation D D DDD such that rk ( D 1 k ¯ ) 2 rk ⁡ D ⊗ 1 k ¯ ≤ 2 rk(D ox1_( bar(k))) <= 2\operatorname{rk}\left(D \otimes 1_{\bar{k}}\right) \leq 2rk⁡(D⊗1k¯)≤2 (Dutta, Gupta, and Lahiri [39]).
Now let R R RRR be a ring containing a field k k kkk. An R R RRR-algebra A A AAA is said to be an A n A n A^(n)\mathbb{A}^{n}An-form over R R RRR with respect to k k kkk if A k k ¯ = ( R k k ¯ ) [ n ] A ⊗ k k ¯ = R ⊗ k k ¯ [ n ] Aox_(k) bar(k)=(Rox_(k)( bar(k)))^([n])A \otimes_{k} \bar{k}=\left(R \otimes_{k} \bar{k}\right)^{[n]}A⊗kk¯=(R⊗kk¯)[n], where k ¯ k ¯ bar(k)\bar{k}k¯ denotes the algebraic closure of k k kkk. A few results on triviality of separable A n A n A^(n)A^{n}An-forms over a ring R R RRR are listed below.
Let A A AAA be an A n A n A^(n)\mathbb{A}^{n}An-form over a ring R R RRR containing a field k k kkk of characteristic 0 . Then:
(1) If n = 1 n = 1 n=1n=1n=1, then A A AAA is isomorphic to the symmetric algebra of a finitely generated rank one projective module over R R RRR [35, THEOREM 7].
(2) If n = 2 n = 2 n=2n=2n=2 and R R RRR is a PID containing Q Q Q\mathbb{Q}Q, then A = R [ 2 ] A = R [ 2 ] A=R^([2])A=R^{[2]}A=R[2] [35, REMARK 8].
(3) If n = 2 n = 2 n=2n=2n=2, then A A AAA is an A 2 A 2 A^(2)\mathbb{A}^{2}A2-fibration over R R RRR.
(4) If n = 2 n = 2 n=2n=2n=2 and R R RRR is a one-dimensional Noetherian domain, then there exists a finitely generated rank-one projective R R RRR-module Q Q QQQ such that A ( Sym R ( Q ) ) [ 1 ] A ≅ Sym R ⁡ ( Q ) [ 1 ] A~=(Sym_(R)(Q))^([1])A \cong\left(\operatorname{Sym}_{R}(Q)\right)^{[1]}A≅(SymR⁡(Q))[1] [39, THEOREM 3.7].
(5) If n = 2 n = 2 n=2n=2n=2 and A A AAA admits has a fixed point free locally nilpotent R R RRR-derivation over any ring R R RRR, then there exists a finitely generated rank one projective R R RRR-module Q Q QQQ such that A ( Sym R ( Q ) ) [ 1 ] A ≅ Sym R ⁡ ( Q ) [ 1 ] A~=(Sym_(R)(Q))^([1])A \cong\left(\operatorname{Sym}_{R}(Q)\right)^{[1]}A≅(SymR⁡(Q))[1] [39, THEOREM 3.8].
The result (3) above shows that an affirmative answer to the A 2 A 2 A^(2)\mathbb{A}^{2}A2-fibration problem (Question 3 (i)) will ensure an affirmative answer to the problem of A 2 A 2 A^(2)\mathbb{A}^{2}A2-forms over general rings. Over a field F F FFF of any characteristic, Das has shown [27] that any factorial A 1 A 1 A^(1)\mathbb{A}^{1}A1-form A A AAA over a ring R R RRR containing F F FFF is trivial if there exists a retraction map from A A AAA to R R RRR.
We cannot say much about A 3 A 3 A^(3)\mathbb{A}^{3}A3-forms over general rings till the time we solve it over fields.

7. AN EXAMPLE OF BHATWADEKAR AND DUTTA

The following example arose from the study of linear planes over a discrete valuation ring by Bhatwadekar and Dutta [12]. Question 5 stated below is an open problem for at least three decades. Let
A = C [ T , X , Y , Z ] and R = C [ T , F ] A A = C [ T , X , Y , Z ]  and  R = C [ T , F ] ⊂ A A=C[T,X,Y,Z]quad" and "quad R=C[T,F]sub AA=\mathbb{C}[T, X, Y, Z] \quad \text { and } \quad R=\mathbb{C}[T, F] \subset AA=C[T,X,Y,Z] and R=C[T,F]⊂A
where F = T X 2 Z + X + T 2 Y + T X Y 2 F = T X 2 Z + X + T 2 Y + T X Y 2 F=TX^(2)Z+X+T^(2)Y+TXY^(2)F=T X^{2} Z+X+T^{2} Y+T X Y^{2}F=TX2Z+X+T2Y+TXY2.
Let
P := X Z + Y 2 G := T Y + X P P := X Z + Y 2 G := T Y + X P {:[P:=XZ+Y^(2)],[G:=TY+XP]:}\begin{aligned} & P:=X Z+Y^{2} \\ & G:=T Y+X P \end{aligned}P:=XZ+Y2G:=TY+XP
and
H := T 2 Z 2 T Y P X P 2 H := T 2 Z − 2 T Y P − X P 2 H:=T^(2)Z-2TYP-XP^(2)H:=T^{2} Z-2 T Y P-X P^{2}H:=T2Z−2TYP−XP2
Then, we can see that
X H + G 2 = T 2 P X H + G 2 = T 2 P XH+G^(2)=T^(2)PX H+G^{2}=T^{2} PXH+G2=T2P
and F = X + T G F = X + T G F=X+TGF=X+T GF=X+TG. Clearly, C [ T , T 1 ] [ F , G , H ] C [ T , T 1 ] [ X , Y , Z ] C T , T − 1 [ F , G , H ] ⊆ C T , T − 1 [ X , Y , Z ] C[T,T^(-1)][F,G,H]subeC[T,T^(-1)][X,Y,Z]\mathbb{C}\left[T, T^{-1}\right][F, G, H] \subseteq \mathbb{C}\left[T, T^{-1}\right][X, Y, Z]C[T,T−1][F,G,H]⊆C[T,T−1][X,Y,Z].
Then the following statements hold:
(i) C [ T , T 1 ] [ X , Y , Z ] = C [ T , T 1 , F , G , H ] = C [ T , T 1 ] [ F ] [ 2 ] C T , T − 1 [ X , Y , Z ] = C T , T − 1 , F , G , H = C T , T − 1 [ F ] [ 2 ] C[T,T^(-1)][X,Y,Z]=C[T,T^(-1),F,G,H]=C[T,T^(-1)][F]^([2])\mathbb{C}\left[T, T^{-1}\right][X, Y, Z]=\mathbb{C}\left[T, T^{-1}, F, G, H\right]=\mathbb{C}\left[T, T^{-1}\right][F]^{[2]}C[T,T−1][X,Y,Z]=C[T,T−1,F,G,H]=C[T,T−1][F][2].
(ii) C [ T , X , Y , Z ] C [ T , X , Y , Z ] C[T,X,Y,Z]\mathbb{C}[T, X, Y, Z]C[T,X,Y,Z] is an A 2 A 2 A^(2)\mathbb{A}^{2}A2-fibration over C [ T , F ] C [ T , F ] C[T,F]\mathbb{C}[T, F]C[T,F].
(iii) C [ T , X , Y , Z ] [ 1 ] = C [ T , F ] [ 3 ] C [ T , X , Y , Z ] [ 1 ] = C [ T , F ] [ 3 ] C[T,X,Y,Z]^([1])=C[T,F]^([3])\mathbb{C}[T, X, Y, Z]^{[1]}=\mathbb{C}[T, F]^{[3]}C[T,X,Y,Z][1]=C[T,F][3]
(iv) C [ T , X , Y , Z ] / ( F ) = C [ T ] [ 2 ] = C [ 3 ] C [ T , X , Y , Z ] / ( F ) = C [ T ] [ 2 ] = C [ 3 ] C[T,X,Y,Z]//(F)=C[T]^([2])=C^([3])\mathbb{C}[T, X, Y, Z] /(F)=\mathbb{C}[T]^{[2]}=\mathbb{C}^{[3]}C[T,X,Y,Z]/(F)=C[T][2]=C[3].
(v) C [ T , X , Y , Z ] / ( F f ( T ) ) = C [ T ] [ 2 ] C [ T , X , Y , Z ] / ( F − f ( T ) ) = C [ T ] [ 2 ] C[T,X,Y,Z]//(F-f(T))=C[T]^([2])\mathbb{C}[T, X, Y, Z] /(F-f(T))=\mathbb{C}[T]^{[2]}C[T,X,Y,Z]/(F−f(T))=C[T][2] for every polynomial f ( T ) C [ T ] f ( T ) ∈ C [ T ] f(T)inC[T]f(T) \in \mathbb{C}[T]f(T)∈C[T].
(vi) C [ T , X , Y , Z ] [ 1 / F ] = C [ T , F , 1 / F , G ] [ 1 ] C [ T , X , Y , Z ] [ 1 / F ] = C [ T , F , 1 / F , G ] [ 1 ] C[T,X,Y,Z][1//F]=C[T,F,1//F,G]^([1])\mathbb{C}[T, X, Y, Z][1 / F]=\mathbb{C}[T, F, 1 / F, G]^{[1]}C[T,X,Y,Z][1/F]=C[T,F,1/F,G][1].
(vii) For any u ( T , F ) R , A [ 1 / u ] = R [ 1 / u ] [ 2 ] u ∈ ( T , F ) R , A [ 1 / u ] = R [ 1 / u ] [ 2 ] u in(T,F)R,A[1//u]=R[1//u]^([2])u \in(T, F) R, A[1 / u]=R[1 / u]^{[2]}u∈(T,F)R,A[1/u]=R[1/u][2], i.e., C [ T , X , Y , Z ] [ 1 / u ] = C [ T , X , Y , Z ] [ 1 / u ] = C[T,X,Y,Z][1//u]=\mathbb{C}[T, X, Y, Z][1 / u]=C[T,X,Y,Z][1/u]= C [ T , F , 1 / u ] [ 2 ] C [ T , F , 1 / u ] [ 2 ] C[T,F,1//u]^([2])\mathbb{C}[T, F, 1 / u]^{[2]}C[T,F,1/u][2].
Question 5.
(a) Is A = C [ T , F ] [ 2 ] ( = R [ 2 ] ) A = C [ T , F ] [ 2 ] = R [ 2 ] A=C[T,F]^([2])(=R^([2]))A=\mathbb{C}[T, F]^{[2]}\left(=R^{[2]}\right)A=C[T,F][2](=R[2]) ?
(b) At least is A = C [ F ] [ 3 ] A = C [ F ] [ 3 ] A=C[F]^([3])A=\mathbb{C}[F]^{[3]}A=C[F][3] ?
If the answer is "No" to (a), then it is a counterexample to the following problems:
(1) A 2 A 2 A^(2)\mathbb{A}^{2}A2-fibration Problem over C [ 2 ] C [ 2 ] C^([2])\mathbb{C}^{[2]}C[2] by (ii).
(2) Cancellation Problem over C [ 2 ] C [ 2 ] C^([2])\mathbb{C}^{[2]}C[2] by (iii).
(3) Epimorphism problem over the ring C [ T ] C [ T ] C[T]\mathbb{C}[T]C[T] (see Question 4 4 ′ 4^(')4^{\prime}4′ ) by (iv).
If the answer is "No" to (b) and hence to (a), then it is a counterexample also to the Epimorphism Problem for C [ 4 ] C [ 3 ] C [ 4 ] → C [ 3 ] C^([4])rarrC^([3])\mathbb{C}^{[4]} \rightarrow \mathbb{C}^{[3]}C[4]→C[3].
Though the above properties have been proved in several places, a proof is presented below. A variant of the Bhatwadekar-Dutta example was also constructed by Vénéreau in his thesis [88]; for a discussion on this and related examples, see [24, 41,64].
Proof. (i) We show that
(1) C [ T , T 1 ] [ X , Y , Z ] = C [ T , T 1 ] [ F , G , H ] (1) C T , T − 1 [ X , Y , Z ] = C T , T − 1 [ F , G , H ] {:(1)C[T,T^(-1)][X","Y","Z]=C[T,T^(-1)][F","G","H]:}\begin{equation*} \mathbb{C}\left[T, T^{-1}\right][X, Y, Z]=\mathbb{C}\left[T, T^{-1}\right][F, G, H] \tag{1} \end{equation*}(1)C[T,T−1][X,Y,Z]=C[T,T−1][F,G,H]
Note that
X = F T G , P = X H + G 2 T 2 Y = ( G X P ) / T X = F − T G , P = X H + G 2 T 2 Y = ( G − X P ) / T {:[X=F-TG","quad P=(XH+G^(2))/(T^(2))],[Y=(G-XP)//T]:}\begin{aligned} & X=F-T G, \quad P=\frac{X H+G^{2}}{T^{2}} \\ & Y=(G-X P) / T \end{aligned}X=F−TG,P=XH+G2T2Y=(G−XP)/T
and
Z = ( H + 2 T Y P + X P 2 ) / T 2 Z = H + 2 T Y P + X P 2 / T 2 Z=(H+2TYP+XP^(2))//T^(2)Z=\left(H+2 T Y P+X P^{2}\right) / T^{2}Z=(H+2TYP+XP2)/T2
and hence equation (1) follows.
(ii) Clearly, A A AAA is a finitely generated R R RRR-algebra. It can be shown by standard arguments that A A AAA is a flat R R RRR-algebra [66, THEOREM 20.H]. We now show that A R k ( p ) = k ( p ) [ 2 ] A ⊗ R k ( p ) = k ( p ) [ 2 ] Aox_(R)k(p)=k(p)^([2])A \otimes_{R} k(p)=k(p)^{[2]}A⊗Rk(p)=k(p)[2] for every prime ideal p p ppp of R R RRR. We note that F X T A F − X ∈ T A F-X in TAF-X \in T AF−X∈TA and hence the image of F F FFF in A / T A A / T A A//TAA / T AA/TA is same as that of X X XXX. Now let p p ppp be a prime ideal of R R RRR. Then either T p T ∈ p T in pT \in pT∈p or T p T ∉ p T!in pT \notin pT∉p. If T p T ∈ p T in pT \in pT∈p, then A R k ( p ) = k ( p ) [ Y , Z ] = k ( p ) [ 2 ] A ⊗ R k ( p ) = k ( p ) [ Y , Z ] = k ( p ) [ 2 ] Aox_(R)k(p)=k(p)[Y,Z]=k(p)^([2])A \otimes_{R} k(p)=k(p)[Y, Z]=k(p)^{[2]}A⊗Rk(p)=k(p)[Y,Z]=k(p)[2]. If T p T ∉ p T!in pT \notin pT∉p, then image of T T TTT in k ( p ) k ( p ) k(p)k(p)k(p) is a unit and the result follows from (i).
(iii) Let D = A [ W ] = C [ T , X , Y , Z , W ] = C [ 5 ] D = A [ W ] = C [ T , X , Y , Z , W ] = C [ 5 ] D=A[W]=C[T,X,Y,Z,W]=C^([5])D=A[W]=\mathbb{C}[T, X, Y, Z, W]=\mathbb{C}^{[5]}D=A[W]=C[T,X,Y,Z,W]=C[5]. We shall show that
D = C [ T , F ] [ 3 ] = R [ 3 ] D = C [ T , F ] [ 3 ] = R [ 3 ] D=C[T,F]^([3])=R^([3])D=\mathbb{C}[T, F]^{[3]}=R^{[3]}D=C[T,F][3]=R[3]
Let
W 1 := T W + P , G 1 := ( G F W 1 ) T = Y X W ( T Y + X P ) ( T W + P ) = Y X W G W 1 , H 1 := { H + 2 G W 1 ( F G T ) W 1 2 } T 2 = Z + 2 Y W X W 2 . W 1 := T W + P , G 1 := G − F W 1 T = Y − X W − ( T Y + X P ) ( T W + P ) = Y − X W − G W 1 , H 1 := H + 2 G W 1 − ( F − G T ) W 1 2 T 2 = Z + 2 Y W − X W 2 . {:[W_(1):=TW+P","],[G_(1):=((G-FW_(1)))/(T)=Y-XW-(TY+XP)(TW+P)=Y-XW-GW_(1)","],[H_(1):=({H+2GW_(1)-(F-GT)W_(1)^(2)})/(T^(2))=Z+2YW-XW^(2).]:}\begin{aligned} & W_{1}:=T W+P, \\ & G_{1}:=\frac{\left(G-F W_{1}\right)}{T}=Y-X W-(T Y+X P)(T W+P)=Y-X W-G W_{1}, \\ & H_{1}:=\frac{\left\{H+2 G W_{1}-(F-G T) W_{1}^{2}\right\}}{T^{2}}=Z+2 Y W-X W^{2} . \end{aligned}W1:=TW+P,G1:=(G−FW1)T=Y−XW−(TY+XP)(TW+P)=Y−XW−GW1,H1:={H+2GW1−(F−GT)W12}T2=Z+2YW−XW2.
Now let
G 2 := G 1 + F W 1 2 = ( Y X W ) T W 1 ( Y X W G W 1 ) = Y X W T W 1 G 1 G 2 := G 1 + F W 1 2 = ( Y − X W ) − T W 1 Y − X W − G W 1 = Y − X W − T W 1 G 1 G_(2):=G_(1)+FW_(1)^(2)=(Y-XW)-TW_(1)(Y-XW-GW_(1))=Y-XW-TW_(1)G_(1)G_{2}:=G_{1}+F W_{1}^{2}=(Y-X W)-T W_{1}\left(Y-X W-G W_{1}\right)=Y-X W-T W_{1} G_{1}G2:=G1+FW12=(Y−XW)−TW1(Y−XW−GW1)=Y−XW−TW1G1
and
W 2 := W 1 ( H 1 F + G 2 2 ) T = W + 2 G 1 W 1 ( Y X W ) G H 1 T G 1 2 W 1 2 W 2 := W 1 − H 1 F + G 2 2 T = W + 2 G 1 W 1 ( Y − X W ) − G H 1 − T G 1 2 W 1 2 W_(2):=(W_(1)-(H_(1)F+G_(2)^(2)))/(T)=W+2G_(1)W_(1)(Y-XW)-GH_(1)-TG_(1)^(2)W_(1)^(2)W_{2}:=\frac{W_{1}-\left(H_{1} F+G_{2}^{2}\right)}{T}=W+2 G_{1} W_{1}(Y-X W)-G H_{1}-T G_{1}^{2} W_{1}^{2}W2:=W1−(H1F+G22)T=W+2G1W1(Y−XW)−GH1−TG12W12
Then, it is easy to see that
D [ T 1 ] = C [ T , T 1 ] [ X , Y , Z , W ] = C [ T , T 1 ] [ F , G , H , W 1 ] = C [ T , T 1 ] [ F , G 1 , H 1 , W 1 ] = C [ T , T 1 ] [ F , G 2 , H 1 , W 2 ] D T − 1 = C T , T − 1 [ X , Y , Z , W ] = C T , T − 1 F , G , H , W 1 = C T , T − 1 F , G 1 , H 1 , W 1 = C T , T − 1 F , G 2 , H 1 , W 2 {:[D[T^(-1)]=C[T,T^(-1)][X","Y","Z","W]],[=C[T,T^(-1)][F,G,H,W_(1)]],[=C[T,T^(-1)][F,G_(1),H_(1),W_(1)]],[=C[T,T^(-1)][F,G_(2),H_(1),W_(2)]]:}\begin{aligned} D\left[T^{-1}\right] & =\mathbb{C}\left[T, T^{-1}\right][X, Y, Z, W] \\ & =\mathbb{C}\left[T, T^{-1}\right]\left[F, G, H, W_{1}\right] \\ & =\mathbb{C}\left[T, T^{-1}\right]\left[F, G_{1}, H_{1}, W_{1}\right] \\ & =\mathbb{C}\left[T, T^{-1}\right]\left[F, G_{2}, H_{1}, W_{2}\right] \end{aligned}D[T−1]=C[T,T−1][X,Y,Z,W]=C[T,T−1][F,G,H,W1]=C[T,T−1][F,G1,H1,W1]=C[T,T−1][F,G2,H1,W2]
and that C [ T , F , G 2 , H 1 , W 2 ] D C T , F , G 2 , H 1 , W 2 ⊆ D C[T,F,G_(2),H_(1),W_(2)]sube D\mathbb{C}\left[T, F, G_{2}, H_{1}, W_{2}\right] \subseteq DC[T,F,G2,H1,W2]⊆D. Let D / T D = C [ x , y , z , w ] D / T D = C [ x , y , z , w ] D//TD=C[x,y,z,w]D / T D=\mathbb{C}[x, y, z, w]D/TD=C[x,y,z,w], where x , y , z , w x , y , z , w x,y,z,wx, y, z, wx,y,z,w denote the images of X , Y , Z , W X , Y , Z , W X,Y,Z,WX, Y, Z, WX,Y,Z,W in D / T D D / T D D//TDD / T DD/TD. We now show that D C [ T , F , G 2 , H 1 , W 2 ] D ⊆ C T , F , G 2 , H 1 , W 2 D subeC[T,F,G_(2),H_(1),W_(2)]D \subseteq \mathbb{C}\left[T, F, G_{2}, H_{1}, W_{2}\right]D⊆C[T,F,G2,H1,W2]. For this, it is enough to show that the kernel of the natural map ϕ : C [ T , F , G 2 , H 1 , W 2 ] D / T D Ï• : C T , F , G 2 , H 1 , W 2 → D / T D phi:C[T,F,G_(2),H_(1),W_(2)]rarr D//TD\phi: \mathbb{C}\left[T, F, G_{2}, H_{1}, W_{2}\right] \rightarrow D / T DÏ•:C[T,F,G2,H1,W2]→D/TD is generated by T T TTT. We note that the image of ϕ Ï• phi\phiÏ• is
C [ x , y x w , z + 2 y w x w 2 , w + 2 p ( y x w x p 2 ) ( y x w ) x p ( z + 2 y w x w 2 ) ] C x , y − x w , z + 2 y w − x w 2 , w + 2 p y − x w − x p 2 ( y − x w ) − x p z + 2 y w − x w 2 C[x,y-xw,z+2yw-xw^(2),w+2p(y-xw-xp^(2))(y-xw)-xp(z+2yw-xw^(2))]\mathbb{C}\left[x, y-x w, z+2 y w-x w^{2}, w+2 p\left(y-x w-x p^{2}\right)(y-x w)-x p\left(z+2 y w-x w^{2}\right)\right]C[x,y−xw,z+2yw−xw2,w+2p(y−xw−xp2)(y−xw)−xp(z+2yw−xw2)],
which is of transcendence degree 4 over C C C\mathbb{C}C. Hence the kernel of ϕ Ï• phi\phiÏ• is a prime ideal of height one and is generated by T T TTT. Therefore, D = C [ T , F , G 2 , H 1 , W 2 ] D = C T , F , G 2 , H 1 , W 2 D=C[T,F,G_(2),H_(1),W_(2)]D=\mathbb{C}\left[T, F, G_{2}, H_{1}, W_{2}\right]D=C[T,F,G2,H1,W2].
(iv)-(v) Let B = C [ T , X , Y , Z ] / ( F f ( T ) ) B = C [ T , X , Y , Z ] / ( F − f ( T ) ) B=C[T,X,Y,Z]//(F-f(T))B=\mathbb{C}[T, X, Y, Z] /(F-f(T))B=C[T,X,Y,Z]/(F−f(T)) for some polynomial f C [ T ] f ∈ C [ T ] f inC[T]f \in \mathbb{C}[T]f∈C[T] and S = C [ T ] S = C [ T ] S=C[T]S=\mathbb{C}[T]S=C[T]. By (ii), it follows that B B BBB is an A 2 A 2 A^(2)\mathbb{A}^{2}A2-fibration over S S SSS. Hence, by Sathaye's theorem [81], B B BBB is locally a polynomial ring over S S SSS and hence by Theorem 4.1, B B BBB is a polynomial ring over S S SSS.
(vi) Let H 1 := F H + G 2 T H 1 := F H + G 2 T H_(1):=(FH+G^(2))/(T)H_{1}:=\frac{F H+G^{2}}{T}H1:=FH+G2T. Then
H 1 = ( X + T G ) ( T 2 Z 2 T Y P X P 2 ) + ( T Y + X P ) 2 T = T P + G H H 1 = ( X + T G ) T 2 Z − 2 T Y P − X P 2 + ( T Y + X P ) 2 T = T P + G H H_(1)=((X+TG)(T^(2)Z-2TYP-XP^(2))+(TY+XP)^(2))/(T)=TP+GHH_{1}=\frac{(X+T G)\left(T^{2} Z-2 T Y P-X P^{2}\right)+(T Y+X P)^{2}}{T}=T P+G HH1=(X+TG)(T2Z−2TYP−XP2)+(TY+XP)2T=TP+GH
Let H 2 := F H 1 + G 3 T H 2 := F H 1 + G 3 T H_(2):=(FH_(1)+G^(3))/(T)H_{2}:=\frac{F H_{1}+G^{3}}{T}H2:=FH1+G3T. Then
H 2 = ( X + T G ) ( T P + G H ) + G 3 T = T ( G 2 H + T G P + X P ) + G ( X H + G 2 ) T = T ( G 2 H + T G P + X P ) + G T 2 P T = G 2 H + X P + 2 T G P H 2 = ( X + T G ) ( T P + G H ) + G 3 T = T G 2 H + T G P + X P + G X H + G 2 T = T G 2 H + T G P + X P + G T 2 P T = G 2 H + X P + 2 T G P {:[H_(2)=((X+TG)(TP+GH)+G^(3))/(T)],[=(T(G^(2)H+TGP+XP)+G(XH+G^(2)))/(T)],[=(T(G^(2)H+TGP+XP)+GT^(2)P)/(T)],[=G^(2)H+XP+2TGP]:}\begin{aligned} H_{2} & =\frac{(X+T G)(T P+G H)+G^{3}}{T} \\ & =\frac{T\left(G^{2} H+T G P+X P\right)+G\left(X H+G^{2}\right)}{T} \\ & =\frac{T\left(G^{2} H+T G P+X P\right)+G T^{2} P}{T} \\ & =G^{2} H+X P+2 T G P \end{aligned}H2=(X+TG)(TP+GH)+G3T=T(G2H+TGP+XP)+G(XH+G2)T=T(G2H+TGP+XP)+GT2PT=G2H+XP+2TGP
Let H 3 := F ( H 2 G ) + G 4 T H 3 := F H 2 − G + G 4 T H_(3):=(F(H_(2)-G)+G^(4))/(T)H_{3}:=\frac{F\left(H_{2}-G\right)+G^{4}}{T}H3:=F(H2−G)+G4T. Then
H 3 = F ( G 2 H + X P + 2 T G P X P T Y ) + G 4 T = F ( 2 T G P T Y ) + G 2 ( F H + G 2 ) T = T F ( 2 G P Y ) + T H 1 G 2 T = F ( 2 G P Y ) + H 1 G 2 H 3 = F G 2 H + X P + 2 T G P − X P − T Y + G 4 T = F ( 2 T G P − T Y ) + G 2 F H + G 2 T = T F ( 2 G P − Y ) + T H 1 G 2 T = F ( 2 G P − Y ) + H 1 G 2 {:[H_(3)=(F(G^(2)H+XP+2TGP-XP-TY)+G^(4))/(T)],[=(F(2TGP-TY)+G^(2)(FH+G^(2)))/(T)],[=(TF(2GP-Y)+TH_(1)G^(2))/(T)],[=F(2GP-Y)+H_(1)G^(2)]:}\begin{aligned} H_{3} & =\frac{F\left(G^{2} H+X P+2 T G P-X P-T Y\right)+G^{4}}{T} \\ & =\frac{F(2 T G P-T Y)+G^{2}\left(F H+G^{2}\right)}{T} \\ & =\frac{T F(2 G P-Y)+T H_{1} G^{2}}{T} \\ & =F(2 G P-Y)+H_{1} G^{2} \end{aligned}H3=F(G2H+XP+2TGP−XP−TY)+G4T=F(2TGP−TY)+G2(FH+G2)T=TF(2GP−Y)+TH1G2T=F(2GP−Y)+H1G2
Now it is easy to see that
C [ T , X , Y , Z , F 1 ] [ T 1 ] = C [ T , T 1 ] [ F , F 1 , G , H ] = C [ T , T 1 ] [ F , F 1 , G , H 1 ] = C [ T , T 1 ] [ F , F 1 , G , H 2 ] = C [ T , T 1 ] [ F , F 1 , G , H 3 ] C T , X , Y , Z , F − 1 T − 1 = C T , T − 1 F , F − 1 , G , H = C T , T − 1 F , F − 1 , G , H 1 = C T , T − 1 F , F − 1 , G , H 2 = C T , T − 1 F , F − 1 , G , H 3 {:[C[T,X,Y,Z,F^(-1)][T^(-1)]=C[T,T^(-1)][F,F^(-1),G,H]],[=C[T,T^(-1)][F,F^(-1),G,H_(1)]],[=C[T,T^(-1)][F,F^(-1),G,H_(2)]],[=C[T,T^(-1)][F,F^(-1),G,H_(3)]]:}\begin{aligned} \mathbb{C}\left[T, X, Y, Z, F^{-1}\right]\left[T^{-1}\right] & =\mathbb{C}\left[T, T^{-1}\right]\left[F, F^{-1}, G, H\right] \\ & =\mathbb{C}\left[T, T^{-1}\right]\left[F, F^{-1}, G, H_{1}\right] \\ & =\mathbb{C}\left[T, T^{-1}\right]\left[F, F^{-1}, G, H_{2}\right] \\ & =\mathbb{C}\left[T, T^{-1}\right]\left[F, F^{-1}, G, H_{3}\right] \end{aligned}C[T,X,Y,Z,F−1][T−1]=C[T,T−1][F,F−1,G,H]=C[T,T−1][F,F−1,G,H1]=C[T,T−1][F,F−1,G,H2]=C[T,T−1][F,F−1,G,H3]
and that the image of C [ T , F , F 1 , G , H 2 ] C T , F , F − 1 , G , H 2 C[T,F,F^(-1),G,H_(2)]\mathbb{C}\left[T, F, F^{-1}, G, H_{2}\right]C[T,F,F−1,G,H2] in A [ F 1 ] / T A [ F 1 ] A F − 1 / T A F − 1 A[F^(-1)]//TA[F^(-1)]A\left[F^{-1}\right] / T A\left[F^{-1}\right]A[F−1]/TA[F−1] is of transcendence degree 3 . Hence A [ F 1 ] = C [ T , F , F 1 , G , H 3 ] = C [ T , F , F 1 , G ] [ 1 ] A F − 1 = C T , F , F − 1 , G , H 3 = C T , F , F − 1 , G [ 1 ] A[F^(-1)]=C[T,F,F^(-1),G,H_(3)]=C[T,F,F^(-1),G]^([1])A\left[F^{-1}\right]=\mathbb{C}\left[T, F, F^{-1}, G, H_{3}\right]=\mathbb{C}\left[T, F, F^{-1}, G\right]^{[1]}A[F−1]=C[T,F,F−1,G,H3]=C[T,F,F−1,G][1].
(vii) Let m m mmm be any maximal ideal of R R RRR other than ( T , F ) ( T , F ) (T,F)(T, F)(T,F). Then either T m T ∉ m T!in mT \notin mT∉m or F m F ∉ m F!in mF \notin mF∉m. Thus, in either case, from (i) and (vi), we have A m = R m [ 2 ] A m = R m [ 2 ] A_(m)=R_(m)^([2])A_{m}=R_{m}^{[2]}Am=Rm[2].
Let u ( T , F ) R u ∈ ( T , F ) R u in(T,F)Ru \in(T, F) Ru∈(T,F)R. Then a maximal ideal of R [ 1 / u ] R [ 1 / u ] R[1//u]R[1 / u]R[1/u] is an extension of a maximal ideal of R R RRR other than ( T , F ) R ( T , F ) R (T,F)R(T, F) R(T,F)R. Hence A [ 1 / u ] A [ 1 / u ] A[1//u]A[1 / u]A[1/u] is a locally polynomial ring in two variables over R [ 1 / u ] R [ 1 / u ] R[1//u]R[1 / u]R[1/u]. Further any projective module over R [ 1 / u ] R [ 1 / u ] R[1//u]R[1 / u]R[1/u] is free. Thus, by Theorem 4.1 , we have A [ 1 / u ] = R [ 1 / u ] [ 2 ] A [ 1 / u ] = R [ 1 / u ] [ 2 ] A[1//u]=R[1//u]^([2])A[1 / u]=R[1 / u]^{[2]}A[1/u]=R[1/u][2].

ACKNOWLEDGMENTS

The author thanks Professor Amartya Kumar Dutta for introducing and guiding her to this world of affine algebraic geometry. The author also thanks him for carefully going through this draft and improving the exposition.

REFERENCES

[1] A. M. Abhyankar and S. M. Bhatwadekar, Generically Laurent polynomial algebras over a D.V.R. which are not quasi Laurent polynomial algebras. J. Pure Appl. Algebra 218 (2014), no. 4, 651-660.
[2] A. M. Abhyankar and S. M. Bhatwadekar, A note on quasi Laurent polynomial algebras in n n nnn variables. J. Commut. Algebra 6 (2014), no. 2, 127-147.
[3] S. Abhyankar, P. Eakin, and W. Heinzer, On the uniqueness of the coefficient ring in a polynomial ring. J. Algebra 23 (1972), 310-342.
[4] S. S. Abhyankar, Polynomials and power series. In Algebra, Arithmetic and Geometry with Applications, edited by C. Christensen et al., pp. 783-784, Springer, 2004.
[5] S. S. Abhyankar and T. T. Moh, Embeddings of the line in the plane. J. Reine Angew. Math. 276 (1975), 148-166.
[6] T. Asanuma, Polynomial fibre rings of algebras over Noetherian rings. Invent. Math. 87 (1987), 101-127.
[7] T. Asanuma, Non-linearizable algebraic k k ∗ k^(**)k^{*}k∗-actions on affine spaces. Invent. Math. 138 (1999), no. 2, 281-306.
[8] T. Asanuma, Purely inseparable k k kkk-forms of affine algebraic curves. In Affine algebraic geometry, pp. 31-46, Contemp. Math. 369, Amer. Math. Soc., Providence, RI, 2005.
[9] T. Asanuma and N. Gupta, On 2-stably isomorphic four dimensional affine domains. J. Commut. Algebra 10 (2018), no. 2, 153-162.
[10] H. Bass, E. H. Connell, and D. L. Wright, Locally polynomial algebras are symmetric algebras. Invent. Math. 38 (1977), 279-299.
[11] S. M. Bhatwadekar, Generalized epimorphism theorem. Proc. Indian Acad. Sci. 98 (1988), no. 2-3, 109-166.
[12] S. M. Bhatwadekar and A. K. Dutta, On affine fibrations. In Commutative algebra: Conf. Comm. Alg. ICTP (1992), edited by A. Simis, N. V. Trung, and G. Valla, pp. 1-17, World Sc., 1994.
[13] S. M. Bhatwadekar and A. K. Dutta, Linear planes over a discrete valuation ring. J. Algebra 166 (1994), no. 2, 393-405.
[14] S. M. Bhatwadekar and A. K. Dutta, On A 1 A 1 A^(1)\mathbb{A}^{1}A1-fibrations of subalgebras of polynomial algebras. Compos. Math. 95 (1995), no. 3, 263-285.
[15] S. M. Bhatwadekar and A. K. Dutta, Structure of A A ∗ A^(**)\mathbb{A}^{*}A∗-fibrations over one-dimensional seminormal semilocal domains. J. Algebra 220 (1999), 561-573.
[16] S. M. Bhatwadekar and A. K. Dutta, On A A ∗ A^(**)\mathbb{A}^{*}A∗-fibrations. J. Pure Appl. Algebra 149 (2000), 1-14.
[17] S. M. Bhatwadekar, A. K. Dutta, and N. Onoda, On algebras which are locally A 1 A 1 A^(1)\mathbb{A}^{1}A1 in codimension-one. Trans. Amer. Math. Soc. 365 (2013), no. 9, 4497-4537.
[18] S. M. Bhatwadekar and N. Gupta, On locally quasi A A ∗ A^(**)\mathbb{A}^{*}A∗ algebras in codimensionone over a Noetherian normal domain. J. Pure Appl. Algebra 215 (2011), 2242-2256.
[19] S. M. Bhatwadekar and N. Gupta, The structure of a Laurent polynomial fibration in n n nnn variables. J. Algebra 353 (2012), no. 1, 142-157.
[20] S. M. Bhatwadekar and N. Gupta, A note on the cancellation property of k [ X , Y ] k [ X , Y ] k[X,Y]k[X, Y]k[X,Y]. J. Algebra Appl. (special issue in honour of Prof. Shreeram S. Abhyankar) 14 (2015), no. 9, 15400071, 5 pp.
[21] S. M. Bhatwadekar and N. Gupta, On quadratic planes. Preprint.
[22] A. J. Crachiola and L. Makar-Limanov, An algebraic proof of a cancellation theorem for surfaces. J. Algebra 3 2 0 3 2 0 320\mathbf{3 2 0}320 (2008), no. 8, 3113-3119.
[23] D. Daigle and G. Freudenburg, Triangular derivations of k [ X 1 , X 2 , X 3 , X 4 ] k X 1 , X 2 , X 3 , X 4 k[X_(1),X_(2),X_(3),X_(4)]k\left[X_{1}, X_{2}, X_{3}, X_{4}\right]k[X1,X2,X3,X4]. J. Algebra 241 (2001), no. 1, 328-339.
[24] D. Daigle and G. Freudenburg, Families of affine fibrations. In Symmetry and spaces, edited by H. E. A. Campbell et al., pp. 35-43, Progr. Math. 278, Birkhäuser, 2010.
[25] D. Daigle and S. Kaliman, A note on locally nilpotent derivations and variables of k [ X , Y , Z ] k [ X , Y , Z ] k[X,Y,Z]k[X, Y, Z]k[X,Y,Z]. Canad. Math. Bull. 52 (2009), no. 4, 535-543.
[26] W. Danielewski, On a Cancellation Problem and automorphism groups of affine algebraic varieties. Preprint, Warsaw, 1989. (Appendix by K. Fieseler).
[27] P. Das, A note on factorial A 1 A 1 A^(1)\mathbb{A}^{1}A1-forms with retractions. Comm. Algebra 40 (2012), no. 9 , 3221 3223 9 , 3221 − 3223 9,3221-32239,3221-32239,3221−3223.
[28] P. Das and A. K. Dutta, Planes of the form b ( X , Y ) Z n a ( X , Y ) b ( X , Y ) Z n − a ( X , Y ) b(X,Y)Z^(n)-a(X,Y)b(X, Y) Z^{n}-a(X, Y)b(X,Y)Zn−a(X,Y) over a DVR. J. Commut. Algebra 3 (2011), no. 4, 491-509.
[29] N. Dasgupta and N. Gupta, An algebraic characterisation of the affine three space. 2019, arXiv:1709.00169. To appear in J. Commut. Algebra. https://projecteuclid. org/journals/jca/journal-of-commutative-algebra/DownloadAcceptedPapers/ 20301.pdf.
[30] A. Dubouloz, The cylinder over the Koras-Russell cubic threefold has a trivial Makar-Limanov invariant. Transform. Groups 14 (2009), no. 3, 531-539.
[31] A. Dubouloz and J. Fasel, Families of A 1 A 1 A^(1)\mathbb{A}^{1}A1-contractible affine threefolds. Algebr. Geom. 5 (2018), no. 1, 1-14.
[32] A. Dubouloz, Affine surfaces with isomorphic A 2 A 2 A^(2)\mathbb{A}^{2}A2-cylinders. Kyoto J. Math. 59 (2019), no. 1, 181-193.
[33] A. Dubouloz, S. Pauli, and P. A. Østvær, A 1 A 1 A^(1)\mathbb{A}^{1}A1-contractibility of affine modifications. Int. J. Math. 14 (2019), no. 30, 1950069.
[34] A. K. Dutta, On A 1 A 1 A^(1)\mathbb{A}^{1}A1-bundles of affine morphisms. J. Math. Kyoto Univ. 35 (1995), no. 3 , 377 385 3 , 377 − 385 3,377-3853,377-3853,377−385.
[35] A. K. Dutta, On separable A 1 A 1 A^(1)\mathbb{A}^{1}A1-forms. Nagoya Math. J. 159 (2000), 45-51.
[36] A. K. Dutta, Some results on affine fibrations. In Advances in algebra and geometry, edited by C. Musili, pp. 7-24, Hindustan Book Agency, India, 2003.
[37] A. K. Dutta, Some results on subalgebras of polynomial algebras. In Commutative algebra and algebraic geometry, edited by S. Ghorpade et al., pp. 85-95 390, Contemp. Math., 2005.
[38] A. K. Dutta and N. Gupta, The epimorphism theorem and its generalisations. J. Algebra Appl. (special issue in honour of Prof. Shreeram S. Abhyankar) 14 (2015), no. 9, 15400010, 30 pp.
[39] A. K. Dutta, N. Gupta, and A. Lahiri, On Separable A 2 A 2 A^(2)\mathbb{A}^{2}A2 and A 3 A 3 A^(3)\mathbb{A}^{3}A3-forms. Nagoya Math. J. 239 (2020), 346-354.
[40] A. K. Dutta and N. Onoda, Some results on codimension-one A 1 A 1 A^(1)\mathbb{A}^{1}A1-fibrations. J. Algebra 313 (2007), 905-921.
[41] G. Freudenburg, Algebraic theory of locally nilpotent derivations. In Invariant Theory and Algebraic Transformation Groups VII, Encyclopaedia Math. Sci. 136, Springer, Berlin, 2006.
[42] G. Freudenburg and P. Russell, Open problems in affine algebraic geometry, In Affine algebraic geometry, pp. 1-30, Contemp. Math. 369, 2005.
[43] T. Fujita, On Zariski problem. Proc. Jpn. Acad. 55 (1979), no. A, 106-110.
[44] N. Gupta, On faithfully flat fibrations by a punctured line. J. Algebra 415 (2014), 13 34 13 − 34 13-3413-3413−34.
[45] N. Gupta, On the Cancellation Problem for the affine space A 3 A 3 A^(3)\mathbb{A}^{3}A3 in characteristic p p ppp. Invent. Math. 195 (2014), 279-288.
[46] N. Gupta, On the family of affine threefolds x m y = F ( x , z , t ) x m y = F ( x , z , t ) x^(m)y=F(x,z,t)x^{m} y=F(x, z, t)xmy=F(x,z,t). Compos. Math. 150 (2014), no. 6, 979-998.
[47] N. Gupta, On Zariski's Cancellation Problem in positive characteristic. Adv. Math. 264 (2014), 296-307.
[48] N. Gupta, A survey on Zariski Cancellation Problem. Indian J. Pure Appl. Math. 46 (2015), no. 6, 865-877.
[49] N. Gupta and S. Sen, On double Danielewski surfaces and the Cancellation Problem. J. Algebra 533 (2019), 25-43.
[50] R. V. Gurjar, A topological proof of cancellation theorem for C 2 C 2 C^(2)\mathbb{C}^{2}C2. Math. Z. 240 (2002), no. 1, 83-94.
[51] R. V. Gurjar, K. Masuda, and M. Miyanishi, Affine space fibrations. In Springer Proceedings in Mathematics and Statistics: Polynomial rings and Affine Algebraic Geometry, (PRAAG) 2018, Tokyo, Japan, February 12-16, pp. 151-194, Springer, 2018.
[52] E. Hamann, On the R R RRR-invariance of R [ x ] R [ x ] R[x]R[x]R[x]. J. Algebra 35 (1975), 1-16.
[53] M. Hochster, Non-uniqueness of the ring of coefficients in a polynomial ring. Proc. Amer. Math. Soc. 34 (1972), no. 1, 81-82.
[54] M. Hoyois, A. Krishna, and P. A. Østvær, A 1 A 1 A^(1)\mathbb{A}^{1}A1-contractibility of Koras-Russell threefolds. Algebr. Geom. 3 (2016), no. 4, 407-423.
[55] S. Kaliman, M. Koras, L. Makar-Limanov, and P. Russell, C C ∗ C^(**)\mathbb{C}^{*}C∗-actions on C 3 C 3 C^(3)\mathbb{C}^{3}C3 are linearizable. Electron. Res. Announc. Am. Math. Soc. 3 (1997), 63-71.
[56] Sh. Kaliman, Polynomials with general C [ 2 ] C [ 2 ] C^([2])\mathbb{C}^{[2]}C[2]-fibers are variables. Pacific J. Math. 203 (2002), no. 1, 161-190.
[57] T. Kambayashi, On the absence of nontrivial separable forms of the affine plane. J. Algebra 35 (1975), 449-456.
[58] T. Kambayashi, Automorphism group of a polynomial ring and algebraic group actions on affine space. J. Algebra 60 (1979), 439-451.
[59] T. Kambayashi and M. Miyanishi, On flat fibrations by affine line. Illinois J. Math. 22 (1978), no. 4, 662-671.
[60] T. Kambayashi and D. Wright, Flat families of affine lines are affine line bundles. Illinois J. Math. 29 (1985), no. 4, 672-681.
[61] M. Koras and P. Russell, Separable forms of G m G m G_(m)\mathbb{G}_{m}Gm-actions on A k 3 A k 3 A_(k)^(3)\mathbb{A}_{k}^{3}Ak3. Transform. Groups 18 (2013), no. 4, 1155-1163.
[62] H. Kraft, Challenging problems on affine n n nnn-space. Séminaire Bourbaki 8 0 2 8 0 2 802\mathbf{8 0 2}802 (1995), 295-317.
[63] T. Y. Lam, Serre's problem on projective modules. Springer, Berlin-Heidelberg, 2006.
[64] D. Lewis, Vénéreau-type polynomials as potential counterexamples. J. Pure Appl. Algebra 217 (2013), no. 5, 946-957.
[65] L. Makar-Limanov, On the hypersurface x + x 2 y + z 2 + t 3 = 0 x + x 2 y + z 2 + t 3 = 0 x+x^(2)y+z^(2)+t^(3)=0x+x^{2} y+z^{2}+t^{3}=0x+x2y+z2+t3=0 in C 4 C 4 C^(4)\mathbb{C}^{4}C4 or a C 3 C 3 C^(3)\mathbb{C}^{3}C3 like threefold which is not C 3 C 3 C^(3)\mathbb{C}^{3}C3. Israel J. Math. 96 (1996), no. B, 419-429.
[66] H. Matsumura, Commutative algebra. 2nd edn. Benjamin, 1980.
[67] M. Miyanishi, An algebraic characterization of the affine plane. J. Math. Kyoto Univ. 15 (1975), 169-184.
[68] M. Miyanishi, An algebro-topological characterization of the affine space of dimension three. Amer. J. Math. 106 (1984), 1469-1486.
[69] M. Miyanishi, Recent developments in affine algebraic geometry: from the personal viewpoints of the author. In Affine algebraic geometry, pp. 307-378, Osaka Univ. Press, Osaka, 2007.
[70] M. Miyanishi and T. Sugie, Affine surfaces containing cylinderlike open sets. J. Math. Kyoto Univ. 20 (1980), 11-42.
[71] M. Nagata, On automorphism group of k [ X , Y ] k [ X , Y ] k[X,Y]k[X, Y]k[X,Y]. Kyoto Univ. Lect. Math. 5, Kinokuniya, Tokyo, 1972.
[72] C. P. Ramanujam, A topological characterization of the affine plane as an algebraic variety. Ann. of Math. 94 (1971), 69-88.
[73] P. Russell, Simple birational extensions of two dimensional affine rational domains. Compos. Math. 33 (1976), no. 2, 197-208.
[74] P. Russell, On affine-ruled rational surfaces. Math. Ann. 255 (1981), 287-302.
[75] P. Russell, Embedding problems in affine algebraic geometry. In Polynomial automorphisms and related topics, edited by H. Bass et al., pp. 113-135, Pub. House for Sc. and Tech., Hanoi, 2007.
[76] P. Russell and A. Sathaye, On finding and cancelling variables in k [ X , Y , Z ] k [ X , Y , Z ] k[X,Y,Z]k[X, Y, Z]k[X,Y,Z]. J. Algebra 57 (1979), no. 1, 151-166.
[77] P. Russell and A. Sathaye, Forty years of the epimorphism theorem. Eur. Math. Soc. Newsl. 90 (2013), 12-17.
[78] L. M. Sangines Garcia, On quadratic planes. Ph.D. Thesis, McGill Univ., 1983.
[79] A. Sathaye, On linear planes. Proc. Amer. Math. Soc. 56 (1976), 1-7.
[80] A. Sathaye, Generalized Newton-Puiseux expansion and Abhyankar-Moh semigroup theorem. Invent. Math. 74 (1983), no. 1, 149-157.
[81] A. Sathaye, Polynomial ring in two variables over a D.V.R.: a criterion. Invent. Math. 74 (1983), 159-168.
[82] A. Sathaye, An application of generalized Newton Puiseux expansions to a conjecture of D. Daigle and G. Freudenburg. In Algebra, arithmetic and geometry with applications, edited by C. Christensen et al., pp. 687-701, Springer, 2004.
[83] B. Segre, Corrispondenze di Möbius e trasformazioni cremoniane intere. Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 91 (1956/1957), 3-19.
[84] A. R. Shastri, Polynomial representations of knots. Tohoku Math. J. (2) 44 (1992), no. 1, 11-17.
[85] A. A. Suslin, Locally polynomial rings and symmetric algebras. Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 3, 503-515 (Russian).
[86] M. Suzuki, Propriétés topologiques des polynômes de deux variables complexes, et automorphismes algébriques de l'espace C 2 C 2 C^(2)\mathbb{C}^{2}C2. J. Math. Soc. Japan 26 (1974), 241-257.
[87] B. Veřsfečler and I. V. Dolgačev, Unipotent group schemes over integral rings. Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 757-799.
[88] S. Vénéreau, Automorphismes et variables de Íanneau de polynómes A [ y 1 , , y m ] A y 1 , … , y m A[y_(1),dots,y_(m)]A\left[y_{1}, \ldots, y_{m}\right]A[y1,…,ym]. Ph.D. Thesis, Institut Fourier, Grenoble, 2001.
[89] D. Wright, Cancellation of variables of the form b T n a b T n − a bT^(n)-ab T^{n}-abTn−a. J. Algebra 52 (1978), no. 1 , 94 100 1 , 94 − 100 1,94-1001,94-1001,94−100.

NEENA GUPTA

Statistics and Mathematics Unit, Indian Statistical Institute, 203 B.T. Road,
Kolkata 700 108, India, neenag @ isical.ac.in, rnanina@ gmail.com

THE FORMAL MODEL OF SEMI-INFINITE FLAG MANIFOLDS

SYU KATO

ABSTRACT

The formal model of semi-infinite flag manifold is a variant of an affine flag variety that was recognized from the 1980s but not studied extensively until the late 2010s. In this note, we exhibit constructions and ideas appearing in our recent study of the formal model of semi-infinite flag manifold of a simple algebraic group. Our results have some implications to the theory of rational maps from a projective line to partial flag manifolds, and also on the structures of quantum cohomologies and quantum K K KKK-groups of partial flag manifolds.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 14M15; Secondary 14D24, 14N35, 20G44, 22E50, 57T15

KEYWORDS

Semi-infinite flag manifold, quasi-map space, quantum K-group, Kac-Moody group, affine
Lie algebra, global Weyl module

1. INTRODUCTION

Compact complex-analytic spaces that admit homogeneous Lie group actions are quite rare in nature, and their classification reduces into three primitive classes: finite groups, tori, and (partial) flag manifolds. The first have discrete topology and the role of geometric consideration is rather small, in general. The second, particularly those admit polarizations, offer a major subject known as abelian varieties. The third, the (partial) flag manifolds of compact simple Lie groups, are ubiquitous in representation theory of semisimple algebraic groups and quantum groups. By the universal nature of general linear groups, flag manifolds of unitary groups are extensively studied from the geometric perspective.
In representation-theoretic considerations, we usually consider flag manifolds as projective algebraic varieties defined over an algebraically closed field (that form a family over Spec Z Spec ⁡ Z Spec Z\operatorname{Spec} \mathbb{Z}Spec⁡Z ). This definition naturally extends to an arbitrary Kac-Moody setting, but the resulting objects have at least two variants, thin flag varieties and thick flag manifolds (defined by Kac-Peterson [75] and Kashiwara [40], respectively). In case the Kac-Moody group is of affine type, we have a loop realization of the corresponding Kac-Moody group, essentially identifying the corresponding group with the set of k ( ( z ) ) k ( ( z ) ) k((z))\mathbb{k}((z))k((z))-valued points of a simple algebraic group over a field k k k\mathbb{k}k. This motivates us to consider yet other versions of flag manifolds of affine type that can be understood as an enhancement of arc schemes of usual flag manifolds. These are the semi-infinite flag manifolds that originate from the ideas of Lusztig [63, $11] and Drinfeld [22]. Lusztig's original idea is to construct varieties that naturally encode representation theory of simple algebraic groups over finite fields. The Lusztig program (see, e.g., [ 44 , 63 ] [ 44 , 63 ] [44,63][44,63][44,63] ) adds representation theory of quantum groups at roots of unity and representation theory of affine Lie algebras at negative rational levels into the picture, and Feigin-Frenkel [19] put representation theory of affine Lie algebras at the critical level into the picture. The semi-infinite flag manifolds itself have two realizations, that we refer to as the ind-model and the formal model. The geometry of the ind-model of semi-infinite flag manifolds, also known as the space of quasimaps from a projective line to a flag manifold, was studied extensively by Braverman, Finkelberg, Mirković, and their collaborators (see [ 8 , 18 , 21 , 22 ] [ 8 , 18 , 21 , 22 ] [8,18,21,22][8,18,21,22][8,18,21,22] ).
One instance of the ind-model of semi-infinite flag manifold is the space of principal bundles on an algebraic curve equipped with some reduction. This interpretation realizes some portion of the above representation-theoretic expectations [2,31]. The formal model of semi-infinite flag manifolds is expected to add a concrete understanding of related representation-theoretic patterns [ 19 , 22 , 63 ] [ 19 , 22 , 63 ] [19,22,63][19,22,63][19,22,63]. Unfortunately, such an idea needs to be polished as its implementation faces difficulty due to its essential infinite-dimensionality. This forces us to employ affine Grassmannians instead of semi-infinite flag manifolds in some cases (see [ 26 , 30 , 78 ] [ 26 , 30 , 78 ] [26,30,78][26,30,78][26,30,78] ) at the moment, that is possible by some tight connections [ 27 , 70 ] [ 27 , 70 ] [27,70][27,70][27,70].
Meanwhile, it is realized that the semi-infinite flag manifold is a version of the loop space of a flag manifold, and hence it is related to its quantum cohomology [32]. In fact, the ind-model of a semi-infinite flag manifold offers a description of the quantum K K KKK-theoretic J J JJJ-function of a flag manifold [9] that encodes its small quantum K K KKK-group.
In both contexts of the above two paragraphs, the Peterson isomorphism [59, 74], that connects the quantum cohomology of a flag manifold with the homology of an affine Grassmannian, should admit an interpretation using a semi-infinite flag manifold. However, such an interpretation is not known today (though we have Corollary 7.3).
The main goal of this note is to explain a realization of the formal model of semiinfinite flag manifold [46,50,52], that is reminiscent to the classical description of the original flag manifolds. Our realization is supported by recent developments in representation theory of affine Lie algebras [ 14 , 15 , 51 ] [ 14 , 15 , 51 ] [14,15,51][14,15,51][14,15,51], that is also reminiscent to the representation theory of simple Lie algebras. It turns out that the study of the formal model of the semi-infinite flag manifold has implications to the corresponding ind-model [50], as well as the study of quantum K K KKK-groups of partial flag manifolds and the K K KKK-groups of affine Grassmannians [45, 47, 48]. This includes an interpretation (and a proof) of the K K KKK-theoretic analogue of the Peterson isomorphism using semi-infinite flag manifolds (Theorem 8.2).
The results presented here describe the formal model of semi-infinite flag manifolds in a down-to-earth fashion, and also provide first nontrivial conclusions deduced from them. However, we have not yet reached our primary goal to understand representation theory from this perspective in a satisfactory fashion. We hope to improve this situation in the near future.
The organization of this note is as follows: We first recall the construction of flag manifolds that is parallel to our later construction in Section 2. We explain the role of quantum groups in the structure theory of Kac-Moody algebras and exhibit two versions of flag varieties of Kac-Moody groups in Section 3. In Section 4, we exhibit some representation theory of affine Lie algebras. Based on it, we explain our construction of the formal model of semi-infinite flag manifolds in Section 5. This enables us to present our idea on the Frobenius splitting of semi-infinite flag manifolds in Section 6. We explain the connection between its Richardson varieties and quasimap spaces in Section 7, and explain how they fit into the study of quantum cohomology of flag manifolds. We exhibit the K K KKK-theoretic Peterson isomorphism in Section 8. We discuss the functoriality of the quantum K K KKK-groups of partial flag manifolds in Section 9. We finish this note by discussing some perspectives in Section 10.
We assume that every field k k k\mathbb{k}k has characteristic 2 ≠ 2 !=2\neq 2≠2. A variety is some algebraicgeometric object that admits singularity, and a manifold is a variety that is supposed to be smooth in some sense. An algebraic variety is a separated scheme of finite type defined over a field (i.e., our variety is not necessarily irreducible or reduced). We set N := Z 0 N := Z ≥ 0 N:=Z_( >= 0)\mathbb{N}:=\mathbb{Z}_{\geq 0}N:=Z≥0.

2. FLAG MANIFOLDS VIA REPRESENTATION THEORY

Let G G GGG be a simply connected semisimple algebraic group over an algebraically closed field k k k\mathbb{k}k. Let T B T ⊂ B T sub BT \subset BT⊂B be its maximal torus and a Borel subgroup (maximal solvable subgroup). Let W ( = N G ( T ) / T ) W = N G ( T ) / T W(=N_(G)(T)//T)W\left(=N_{G}(T) / T\right)W(=NG(T)/T) be the Weyl group of G G GGG. Let X X X\mathbb{X}X be the set of onedimensional rational T T TTT-characters (the set of T T TTT-weights), that admits a natural W W WWW-action. We set X + := i = 1 r N i X + := ∑ i = 1 r   N i X_(+):=sum_(i=1)^(r)N_(i)\mathbb{X}_{+}:=\sum_{i=1}^{r} \mathbb{N}_{i}X+:=∑i=1rNi, where ϖ 1 , , ϖ r X Ï– 1 , … , Ï– r ∈ X Ï–_(1),dots,Ï–_(r)inX\varpi_{1}, \ldots, \varpi_{r} \in \mathbb{X}Ï–1,…,Ï–r∈X are fundamental weights with respect to B B BBB. The set of isomorphism classes of irreducible rational representations { L ( λ ) } λ { L ( λ ) } λ {L(lambda)}_(lambda)\{L(\lambda)\}_{\lambda}{L(λ)}λ of G G GGG is labeled by X + X + X_(+)\mathbb{X}_{+}X+in such a way that each L ( λ ) L ( λ ) L(lambda)L(\lambda)L(λ) contains a unique (up to scalar) B B BBB-eigenvector
v λ v λ v_(lambda)\mathbf{v}_{\lambda}vλ with its T T TTT-weight λ λ lambda\lambdaλ. We refer λ ( X + ) λ ∈ X + lambda(inX_(+))\lambda\left(\in \mathbb{X}_{+}\right)λ(∈X+)as the highest weight of L ( λ ) L ( λ ) L(lambda)L(\lambda)L(λ). The flag manifold B := G / B B := G / B B:=G//B\mathscr{B}:=G / BB:=G/B of G G GGG is the maximal G G GGG-homogeneous space that is projective.
In case k = C k = C k=C\mathbb{k}=\mathbb{C}k=C, we have
B = ( y E ) / T B = ( y ∖ E ) / T B=(y\\E)//T\mathscr{B}=(y \backslash E) / TB=(y∖E)/T
where y y yyy is an affine algebraic variety with ( G × T ) ( G × T ) (G xx T)(G \times T)(G×T)-action whose ring C [ y ] C [ y ] C[y]\mathbb{C}[y]C[y] of regular functions is written as
(2.1) C [ y ] λ X + L ( λ ) ( as G × T -modules ) (2.1) C [ y ] ≅ λ ∈ X + L ( λ ) ∗ (  as  G × T -modules  ) {:(2.1)C[y]~=_(lambda inX_(+))L(lambda)^(**)quad(" as "G xx T"-modules "):}\begin{equation*} \mathbb{C}[y] \cong \underset{\lambda \in \mathbb{X}_{+}}{ } L(\lambda)^{*} \quad(\text { as } G \times T \text {-modules }) \tag{2.1} \end{equation*}(2.1)C[y]≅λ∈X+L(λ)∗( as G×T-modules )
and E y E ⊂ y E sub yE \subset yE⊂y is the locus where the T T TTT-action is not free. Here, the G G GGG-action on C [ y ] C [ y ] C[y]\mathbb{C}[y]C[y] is the natural actions on L ( λ ) L ( λ ) L(lambda)L(\lambda)L(λ), and the T T TTT-action on C [ y ] C [ y ] C[y]\mathbb{C}[y]C[y] comes from the grading X + X X + ⊂ X X_(+)subX\mathbb{X}_{+} \subset \mathbb{X}X+⊂X in the RHS of (2.1). These data, together with the condition E y E ≠ y E!=yE \neq yE≠y, essentially determine C [ y ] C [ y ] C[y]\mathbb{C}[y]C[y] as C C C\mathbb{C}C-algebras generated by L ( ϖ i ) L Ï– i ∗ L(Ï–_(i))^(**)L\left(\varpi_{i}\right)^{*}L(Ï–i)∗ for 1 i r 1 ≤ i ≤ r 1 <= i <= r1 \leq i \leq r1≤i≤r. Consider a point x 0 y x 0 ∈ y x_(0)in yx_{0} \in yx0∈y given by { v λ } λ v λ λ {v_(lambda)}lambda\left\{\mathbf{v}_{\lambda}\right\} \lambda{vλ}λ, seen as linear maps on { L ( λ ) } λ L ( λ ) ∗ λ {L(lambda)^(**)}_(lambda)\left\{L(\lambda)^{*}\right\}_{\lambda}{L(λ)∗}λ. The image [ x 0 ] x 0 [x_(0)]\left[x_{0}\right][x0] of this point x 0 x 0 x_(0)x_{0}x0 has its G G GGG-stabilizer equal to B B BBB. This induces an inclusion
G / B B i = 1 r P ( L ( ϖ i ) ) G / B ↪ B ⊂ ∏ i = 1 r   P L Ï– i G//B↪Bsubprod_(i=1)^(r)P(L(Ï–_(i)))G / B \hookrightarrow \mathscr{B} \subset \prod_{i=1}^{r} \mathbb{P}\left(L\left(\varpi_{i}\right)\right)G/B↪B⊂∏i=1rP(L(Ï–i))
induced from B / B [ x 0 ] B / B ↦ x 0 B//B|->[x_(0)]B / B \mapsto\left[x_{0}\right]B/B↦[x0] by the G G GGG-action. (One needs additional representation-theoretic analysis to conclude G / B B G / B ≅ B G//B~=BG / B \cong \mathscr{B}G/B≅B.) This consideration transfers all geometric statements relevant to B B B\mathscr{B}B to algebraic statements on the space in (2.1) in principle, but most of the geometric results on B B B\mathscr{B}B and its subvarieties were proved for the first time by other methods (see, e.g., [56]).
Note that the vector space (2.1) does not acquire the structure of a ring when char k = k = k=\mathbb{k}=k= p > 0 p > 0 p > 0p>0p>0. The reason is that we do not have a map L ( λ ) L ( μ ) L ( λ + μ ) L ( λ ) ∗ ⊗ L ( μ ) ∗ → L ( λ + μ ) ∗ L(lambda)^(**)ox L(mu)^(**)rarr L(lambda+mu)^(**)L(\lambda)^{*} \otimes L(\mu)^{*} \rightarrow L(\lambda+\mu)^{*}L(λ)∗⊗L(μ)∗→L(λ+μ)∗, or equivalently, L ( λ + μ ) L ( λ ) L ( μ ) L ( λ + μ ) → L ( λ ) ⊗ L ( μ ) L(lambda+mu)rarr L(lambda)ox L(mu)L(\lambda+\mu) \rightarrow L(\lambda) \otimes L(\mu)L(λ+μ)→L(λ)⊗L(μ) for general λ , μ X + λ , μ ∈ X + lambda,mu inX_(+)\lambda, \mu \in \mathbb{X}_{+}λ,μ∈X+. One way to improve the situation is to replace { L ( λ ) } λ X + { L ( λ ) } λ ∈ X + {L(lambda)}_(lambda inX_(+))\{L(\lambda)\}_{\lambda \in \mathbb{X}_{+}}{L(λ)}λ∈X+with a suitable family of modules { Y ( λ ) } λ X + { Y ( λ ) } λ ∈ X + {Y(lambda)}_(lambda inX_(+))\{Y(\lambda)\}_{\lambda \in \mathbb{X}_{+}}{Y(λ)}λ∈X+with larger members such that the G G GGG-module map
(2.2) Y ( λ + μ ) Y ( λ ) Y ( μ ) (2.2) Y ( λ + μ ) → Y ( λ ) ⊗ Y ( μ ) {:(2.2)Y(lambda+mu)rarr Y(lambda)ox Y(mu):}\begin{equation*} Y(\lambda+\mu) \rightarrow Y(\lambda) \otimes Y(\mu) \tag{2.2} \end{equation*}(2.2)Y(λ+μ)→Y(λ)⊗Y(μ)
exists uniquely (up to constant) for every λ , μ X + λ , μ ∈ X + lambda,mu inX_(+)\lambda, \mu \in \mathbb{X}_{+}λ,μ∈X+. It yields an analogous ring of (2.1) that should be closely related to B B B\mathscr{B}B. A standard choice of Y ( λ ) ( λ X + ) Y ( λ ) λ ∈ X + Y(lambda)(lambda inX_(+))Y(\lambda)\left(\lambda \in \mathbb{X}_{+}\right)Y(λ)(λ∈X+)is the Weyl module V ( λ ) V ( λ ) V(lambda)V(\lambda)V(λ) of G G GGG, that is, the projective cover of L ( λ ) L ( λ ) L(lambda)L(\lambda)L(λ) in the categories of rational G G GGG-modules whose composition factors are in { L ( μ ) } λ μ X + { L ( μ ) } λ ≥ μ ∈ X + {L(mu)}_(lambda >= mu inX_(+))\{L(\mu)\}_{\lambda \geq \mu \in \mathbb{X}_{+}}{L(μ)}λ≥μ∈X+, where ≥ >=\geq≥is the dominance ordering on X X X\mathbb{X}X. This produces B B B\mathscr{B}B for all characteristics.
Theorem 2.1 (Orthogonality of Weyl modules, [36, II $4.13]). For each λ , μ X + λ , μ ∈ X + lambda,mu inX_(+)\lambda, \mu \in \mathbb{X}_{+}λ,μ∈X+, we have
Ext G i ( V ( λ ) , V ( μ ) ) k δ i , 0 δ λ , μ Ext G i ⁡ V ( λ ) , V ( μ ) ∗ ≅ k ⊕ δ i , 0 δ λ , μ ∗ Ext_(G)^(i)(V(lambda),V(mu)^(**))~=k^(o+delta_(i,0)delta_(lambda,mu^(**)))\operatorname{Ext}_{G}^{i}\left(V(\lambda), V(\mu)^{*}\right) \cong \mathbb{k}^{\oplus \delta_{i, 0} \delta_{\lambda, \mu^{*}}}ExtGi⁡(V(λ),V(μ)∗)≅k⊕δi,0δλ,μ∗
where μ μ ∗ mu^(**)\mu^{*}μ∗ is the highest weight of L ( μ ) L ( μ ) ∗ L(mu)^(**)L(\mu)^{*}L(μ)∗. By taking the Euler-Poincaré characteristic, this Ext-orthogonality implies the orthogonality of the T T TTT-characters of V ( λ ) V ( λ ) V(lambda)V(\lambda)V(λ). In particular, the T T TTT-characters of V ( λ ) V ( λ ) V(lambda)V(\lambda)V(λ) do not depend on k k k\mathbb{k}k.
Note that L ( λ ) = V ( λ ) L ( λ ) = V ( λ ) L(lambda)=V(lambda)L(\lambda)=V(\lambda)L(λ)=V(λ) for char k = 0 k = 0 k=0\mathbb{k}=0k=0 by the semisimplicity of representations, and hence Theorem 2.1 is Schur's lemma in such a case. As V ( λ ) = k Z V Z ( λ ) V ( λ ) = k ⊗ Z V Z ( λ ) V(lambda)=kox_(Z)V_(Z)(lambda)V(\lambda)=\mathbb{k} \otimes_{\mathbb{Z}} V_{\mathbb{Z}}(\lambda)V(λ)=k⊗ZVZ(λ) holds for a collection of free Z Z Z\mathbb{Z}Z-modules V Z ( λ ) ( λ X + ) V Z ( λ ) λ ∈ X + V_(Z)(lambda)(lambda inX_(+))V_{\mathbb{Z}}(\lambda)\left(\lambda \in \mathbb{X}_{+}\right)VZ(λ)(λ∈X+), we find that B B B\mathscr{B}B extends to a scheme flat over Z Z Z\mathbb{Z}Z. Another possible choice of Y ( λ ) ( λ X + Y ( λ ) λ ∈ X + Y(lambda)(lambda inX_(+):}Y(\lambda)\left(\lambda \in \mathbb{X}_{+}\right.Y(λ)(λ∈X+), the Verma module M ( λ ) M ( λ ) M(lambda)M(\lambda)M(λ) of the (divided power) enveloping algebra of Lie G G GGG, produces an open dense B B BBB-orbit in B B B\mathscr{B}B.

3. KAC-MOODY FLAG VARIETIES

Let us keep the setting of the previous section.

3.1. Reminder on Kac-Moody algebras and their quantum groups

Let g C g C g_(C)g_{C}gC be the Kac-Moody algebra associated to a symmetrizable generalized Cartan matrix ( = = === GCM) C C CCC (see [38]). In case char k = 0 k = 0 k=0k=0k=0, we have the notion of the highest weight integrable representations of g C g C g_(C)g_{C}gC parametrized by the set of dominant weights P + P + P_(+)P_{+}P+ defined similarly to X + X + X_(+)\mathbb{X}_{+}X+. Let L ( Λ ) L ( Λ ) L(Lambda)L(\Lambda)L(Λ) denote the highest weight integrable representation of g C g C g_(C)\mathrm{g}_{C}gC corresponding to Λ P + Λ ∈ P + Lambda inP_(+)\Lambda \in P_{+}Λ∈P+.
We have the quantum group (or the quantized enveloping algebra) U q ( g C ) U q g C U_(q)(g_(C))U_{q}\left(g_{C}\right)Uq(gC) of g C g C g_(C)g_{C}gC originally defined by Drinfeld and Jimbo in the 1980s [17, 37]. It is an algebra defined over Q ( q ) Q ( q ) Q(q)\mathbb{Q}(q)Q(q), and the specialization q 1 q ↦ 1 q|->1q \mapsto 1q↦1 recovers the universal enveloping algebra U ( g C ) U g C U(g_(C))U\left(\mathrm{~g}_{C}\right)U( gC) of g C g C g_(C)\mathrm{g}_{C}gC. Kashiwara [41] and Lusztig [63] defined the canonical/global bases (of the positive/negative parts U q ± ( g C ) ) U q ± g C {:U_(q)^(+-)(g_(C)))\left.U_{q}^{ \pm}\left(g_{C}\right)\right)Uq±(gC)) of U q ( g C ) U q g C U_(q)(g_(C))U_{q}\left(g_{C}\right)Uq(gC) and their integrable representations that generate their Q [ q ] Q [ q ] − Q[q]-\mathbb{Q}[q]-Q[q]− lattices. The construction of Lusztig [64] clarified that quantum groups are, in fact, defined over Z [ q ± 1 ] Z q ± 1 Z[q^(+-1)]\mathbb{Z}\left[q^{ \pm 1}\right]Z[q±1] (or even over N [ q ± 1 ] N q ± 1 N[q^(+-1)]\mathbb{N}\left[q^{ \pm 1}\right]N[q±1] if one can say). In the 2010s, the categorification theorems of a quantum group and its integrable representations appeared [ 39 , 53 , 76 , 77 ] [ 39 , 53 , 76 , 77 ] [39,53,76,77][39,53,76,77][39,53,76,77], and there every algebra that admits a categorification has a suitable Z [ q ] Z [ q ] Z[q]\mathbb{Z}[q]Z[q]-integral structure with distinguished bases, being the Grothendieck group of a module category of a finitely-generated graded algebras (called KLR algebras or quiver Hecke algebras). Therefore, the following is now widely recognized:
Theorem 3.1 (Lusztig [ 63 , 64 , 66 ] [ 63 , 64 , 66 ] [63,64,66][63,64,66][63,64,66] and Kashiwara [41-43]). Assume that k = C k = C k=C\mathbb{k}=\mathbb{C}k=C. The (lower) global bases of U q ± ( g C ) U q ± g C U_(q)^(+-)(g_(C))U_{q}^{ \pm}\left(g_{C}\right)Uq±(gC) induce a Z Z Z\mathbb{Z}Z-integral form U Z ( g C ) U Z g C U_(Z)(g_(C))U_{\mathbb{Z}}\left(g_{C}\right)UZ(gC) of U ( g C ) U g C U(g_(C))U\left(g_{C}\right)U(gC) via q 1 q ↦ 1 q|->1q \mapsto 1q↦1. For each Λ P + Λ ∈ P + Lambda inP_(+)\Lambda \in P_{+}Λ∈P+, we have a Z Z Z\mathbb{Z}Z-lattice L ( Λ ) Z L ( Λ ) Z L(Lambda)_(Z)L(\Lambda)_{\mathbb{Z}}L(Λ)Z of L ( Λ ) L ( Λ ) L(Lambda)L(\Lambda)L(Λ) obtained from the (lower) global base of the corresponding integrable highest weight module of U q ( g C ) U q g C U_(q)(g_(C))U_{q}\left(g_{C}\right)Uq(gC). In addition, L ( Λ ) Z L ( Λ ) Z L(Lambda)_(Z)L(\Lambda)_{\mathbb{Z}}L(Λ)Z is generated by the U Z ( g C ) U Z g C U_(Z)(g_(C))U_{\mathbb{Z}}\left(\mathrm{g}_{C}\right)UZ(gC)-action from a highest weight vector of L ( Λ ) L ( Λ ) L(Lambda)L(\Lambda)L(Λ).
By a specialization of L ( Λ ) Z L ( Λ ) Z L(Lambda)_(Z)L(\Lambda)_{\mathbb{Z}}L(Λ)Z, we obtain a highest weight integrable module L ( Λ ) L ( Λ ) L(Lambda)L(\Lambda)L(Λ) over an arbitrary field k k k\mathbb{k}k. The module L ( Λ ) L ( Λ ) L(Lambda)L(\Lambda)L(Λ) is no longer irreducible when char k > 0 k > 0 k > 0\mathbb{k}>0k>0 (in general), and hence it is a g C g C g_(C)\mathrm{g}_{C}gC-analogue of Weyl modules rather than L ( λ ) L ( λ ) L(lambda)L(\lambda)L(λ) for G G GGG; it is a lack of brevity of the author to choose this notation here. We close this subsection by noting that the integral forms at the end of Section 2 coincide with the integral forms in Theorem 3.1.

3.2. Thin and thick flag varieties

Presentations of the flag varieties for general Kac-Moody groups E E E\mathscr{E}E associated to a GCM C C CCC are similar to those in the previous section. A triangular decomposition of g C g C g_(C)g_{C}gC yields an analogous group â„“ â„“\ellâ„“ to the Borel subgroup. Let T T T\mathcal{T}T be a (standard) maximal torus of â„“ â„“\ellâ„“. The highest weight vector in L ( Λ ) L ( Λ ) L(Lambda)L(\Lambda)L(Λ) is precisely an â„“ â„“\ellâ„“-eigenvector with its T T T\mathcal{T}T-weight Λ Î› Lambda\LambdaΛ. Therefore, the construction in the previous section produces E / E / â„“ E//â„“\mathscr{E} / \mathscr{\ell}E/â„“ via the ring
(3.1) Λ P + L ( Λ ) Λ P + L ( Λ ) (3.1) ⨁ Λ ∈ P +   L ( Λ ) ∨ ⊂ ⨁ Λ ∈ P +   L ( Λ ) ∗ {:(3.1)bigoplus_(Lambda inP_(+))L(Lambda)^(vv)subbigoplus_(Lambda inP_(+))L(Lambda)^(**):}\begin{equation*} \bigoplus_{\Lambda \in P_{+}} L(\Lambda)^{\vee} \subset \bigoplus_{\Lambda \in P_{+}} L(\Lambda)^{*} \tag{3.1} \end{equation*}(3.1)⨁Λ∈P+L(Λ)∨⊂⨁Λ∈P+L(Λ)∗
where L ( Λ ) L ( Λ ) ∗ L(Lambda)^(**)L(\Lambda)^{*}L(Λ)∗ is the vector space dual of L ( Λ ) L ( Λ ) L(Lambda)L(\Lambda)L(Λ), and L ( Λ ) L ( Λ ) ∨ L(Lambda)^(vv)L(\Lambda)^{\vee}L(Λ)∨ is the restricted dual of L ( Λ ) L ( Λ ) L(Lambda)L(\Lambda)L(Λ), defined to be the direct sum of (finite-dimensional) vector space duals offered by the T T T\mathcal{T}T-weight decomposition of L ( Λ ) L ( Λ ) L(Lambda)L(\Lambda)L(Λ).
In this case, both vector spaces in (3.1) are naturally rings. This corresponds to the choice of E E E\mathscr{E}E. The former ring defines B C thick = E / d [ 40 , 49 , 71 ] B C thick  = E / d [ 40 , 49 , 71 ] B_(C)^("thick ")=E//d[40,49,71]\mathscr{B}_{C}^{\text {thick }}=\mathscr{E} / \mathscr{d}[40,49,71]BCthick =E/d[40,49,71] if we take E E E\mathscr{E}E to be a version of the Kac-Moody group that is completed with respect to the opposite direction to ℓ ℓ\mathscr{\ell}ℓ. (This is the maximal Kac-Moody group, but the completion is taken in the opposite way as in the literature.) The latter ring can be seen as the projective limit of finitely-generated algebras, and the union of the spectrums of these rings yields B C thin = E / L B C thin  = E / L B_(C)^("thin ")=E//L\mathscr{B}_{C}^{\text {thin }}=\mathscr{E} / \mathscr{L}BCthin =E/L [56,75] if we take E E E\mathscr{E}E as the uncompleted Kac-Moody group (the Kac-Peterson group or the minimal Kac-Moody group), or as the maximal Kac-Moody group completed with respect to the direction of ℓ ℓ\ellℓ. In other words, we have variants of flag manifolds of Kac-Moody groups associated to a GCM C C CCC as:
(3.2) n B C , n thin = B C thin B C thick (3.2) ⋃ n   B C , n thin  = B C thin  ⊂ B C thick  {:(3.2)uuu_(n)B_(C,n)^("thin ")=B_(C)^("thin ")subB_(C)^("thick "):}\begin{equation*} \bigcup_{n} \mathscr{B}_{C, n}^{\text {thin }}=\mathscr{B}_{C}^{\text {thin }} \subset \mathscr{B}_{C}^{\text {thick }} \tag{3.2} \end{equation*}(3.2)⋃nBC,nthin =BCthin ⊂BCthick 
The scheme B C thick B C thick  B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }}BCthick  is a union of infinite-dimensional affine spaces, and hence is smooth. However, B C thick B C thick  B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }}BCthick  is not compact in an essential way [24]. This picture is compatible with the fact that the Kac-Peterson group is defined by one-parameter generators (and relations), and hence B C thin B C thin  B_(C)^("thin ")\mathscr{B}_{C}^{\text {thin }}BCthin  is a union of finite-dimensional subvarieties B C , n thin B C , n thin  B_(C,n)^("thin ")\mathscr{B}_{C, n}^{\text {thin }}BC,nthin  consisting of points presented by a product of at most n n nnn generating elements. As such, each scheme B C , n thin B C , n thin  B_(C,n)^("thin ")\mathscr{B}_{C, n}^{\text {thin }}BC,nthin  is singular, and hence B C thin B C thin  B_(C)^("thin ")\mathscr{B}_{C}^{\text {thin }}BCthin  is understood to be singular. In fact, it does not admit an inductive limit description by finite-dimensional smooth pieces [24].

4. GLOBAL WEYL MODULES AND THEIR PROJECTIVITY

Let us consider the untwisted affine Kac-Moody case hereafter, with the same conventions as in the previous sections. In particular, our Kac-Moody groups are extensions of the groups
G ( ( z ) ) := G ( k ( ( z ) ) ) and G [ z ± 1 ] := G ( k [ z ± 1 ] ) G ( ( z ) ) := G ( k ( ( z ) ) )  and  G z ± 1 := G k z ± 1 G((z)):=G(k((z)))quad" and "quad G[z^(+-1)]:=G(k[z^(+-1)])G((z)):=G(\mathbb{k}((z))) \quad \text { and } \quad G\left[z^{ \pm 1}\right]:=G\left(\mathbb{k}\left[z^{ \pm 1}\right]\right)G((z)):=G(k((z))) and G[z±1]:=G(k[z±1])
by the loop rotation G m G m G_(m)\mathbb{G}_{m}Gm-actions (that we denote by G m rot G m rot  G_(m)^("rot ")\mathbb{G}_{m}^{\text {rot }}Gmrot  ) and the central extension G m G m G_(m)\mathbb{G}_{m}Gm actions. (These correspond to the maximal/minimal realizations of the Kac-Moody groups
in the previous section.) These are not (pro-)algebraic groups, and it sometimes causes difficulty. Nevertheless, each rational representation V V VVV of G G GGG induces representations
V ( ( z ) ) := V k k ( ( z ) ) and V [ z ± 1 ] := V k k [ z ± 1 ] V ( ( z ) ) := V ⊗ k k ( ( z ) )  and  V z ± 1 := V ⊗ k k z ± 1 V((z)):=Vox_(k)k((z))quad" and "quad V[z^(+-1)]:=Vox_(k)k[z^(+-1)]V((z)):=V \otimes_{\mathbb{k}} \mathbb{k}((z)) \quad \text { and } \quad V\left[z^{ \pm 1}\right]:=V \otimes_{\mathfrak{k}} \mathbb{k}\left[z^{ \pm 1}\right]V((z)):=V⊗kk((z)) and V[z±1]:=V⊗kk[z±1]
of G ( ( z ) ) G ( ( z ) ) G((z))G((z))G((z)) and G [ z ± 1 ] G z ± 1 G[z^(+-1)]G\left[z^{ \pm 1}\right]G[z±1], respectively. These representations are not of highest weight, but still integrable representations when we lift them to the central extensions of G ( ( z ) ) G ( ( z ) ) G((z))G((z))G((z)) and G [ z ± 1 ] G z ± 1 G[z^(+-1)]G\left[z^{ \pm 1}\right]G[z±1] by letting the center G m G m G_(m)\mathbb{G}_{m}Gm act trivially (i.e., they are level-zero integrable representations viewed as representations of affine Lie algebras).
In addition to the T T TTT-action, the representation V [ z ± 1 ] V z ± 1 V[z^(+-1)]V\left[z^{ \pm 1}\right]V[z±1] carries G m rot G m rot  G_(m)^("rot ")\mathbb{G}_{m}^{\text {rot }}Gmrot -action. Let δ δ delta\deltaδ be the degree-one character of G m rot G m rot  G_(m)^("rot ")\mathbb{G}_{m}^{\text {rot }}Gmrot , and set q := e δ q := e δ q:=e^(delta)q:=e^{\delta}q:=eδ. By abuse of notation, we might consider q n q n q^(n)q^{n}qn ( n Z ) ( n ∈ Z ) (n inZ)(n \in \mathbb{Z})(n∈Z) as the functor that twists the G m rot G m rot  G_(m)^("rot ")\mathbb{G}_{m}^{\text {rot }}Gmrot -action by degree n n nnn. We define a graded character of a semisimple ( T × G m rot ) T × G m rot  (T xxG_(m)^("rot "))\left(T \times \mathbb{G}_{m}^{\text {rot }}\right)(T×Gmrot )-module U U UUU as
gch U := n Z λ X q n e λ dim Hom T × G m r o t ( C λ + n δ , U ) gch ⁡ U := ∑ n ∈ Z   ∑ λ ∈ X   q n e λ dim ⁡ Hom T × G m r o t ⁡ C λ + n δ , U gch U:=sum_(n inZ)sum_(lambda inX)q^(n)e^(lambda)dim Hom_(T xxG_(m)^(rot))(C_(lambda+n delta),U)\operatorname{gch} U:=\sum_{n \in \mathbb{Z}} \sum_{\lambda \in \mathbb{X}} q^{n} e^{\lambda} \operatorname{dim} \operatorname{Hom}_{T \times \mathbb{G}_{m}^{\mathrm{rot}}}\left(\mathbb{C}_{\lambda+n \delta}, U\right)gch⁡U:=∑n∈Z∑λ∈Xqneλdim⁡HomT×Gmrot⁡(Cλ+nδ,U)
Then, gch V [ z ± 1 ] V z ± 1 V[z^(+-1)]V\left[z^{ \pm 1}\right]V[z±1] makes sense as all the coefficients are in Z Z Z\mathbb{Z}Z. However, if we take the second symmetric power S 2 ( V [ z ± 1 ] ) S 2 V z ± 1 S^(2)(V[z^(+-1)])S^{2}\left(V\left[z^{ \pm 1}\right]\right)S2(V[z±1]) of V [ z ± 1 ] V z ± 1 V[z^(+-1)]V\left[z^{ \pm 1}\right]V[z±1] over k k k\mathbb{k}k, then it contains an infinity as a coefficient. To avoid such a complication, we sometimes restrict ourselves to the subgroups
G [ [ z ] ] := G ( k [ [ z ] ] ) G ( ( z ) ) and G [ z ] := G ( k [ z ] ) G [ z ± 1 ] G [ [ z ] ] := G ( k [ [ z ] ] ) ⊂ G ( ( z ) )  and  G [ z ] := G ( k [ z ] ) ⊂ G z ± 1 G[[z]]:=G(k[[z]])sub G((z))quad" and "quad G[z]:=G(k[z])sub G[z^(+-1)]G \llbracket z \rrbracket:=G(\mathbb{k} \llbracket z \rrbracket) \subset G((z)) \quad \text { and } \quad G[z]:=G(\mathbb{k}[z]) \subset G\left[z^{ \pm 1}\right]G[[z]]:=G(k[[z]])⊂G((z)) and G[z]:=G(k[z])⊂G[z±1]
We sometimes use the subgroup I G [ [ z ] ] I ⊂ G [ [ z ] ] Isub G[[z]]\mathbf{I} \subset G \llbracket z \rrbracketI⊂G[[z]] defined by the pullback of B B BBB under the evaluation map e v 0 : G [ [ z ] ] G e v 0 : G [ [ z ] ] → G ev_(0):G[[z]]rarr G\mathrm{ev}_{0}: G \llbracket z \rrbracket \rightarrow Gev0:G[[z]]→G at z = 0 z = 0 z=0z=0z=0. The group I I I\mathbf{I}I is the Iwahori subgroup obtained from (the completed version of ) d ) d )d) d)d by removing G m rot G m rot  G_(m)^("rot ")\mathbb{G}_{m}^{\text {rot }}Gmrot  and quotient out by the central extension.
By the quotient map k [ z ] k k [ z ] → k k[z]rarrk\mathbb{k}[z] \rightarrow \mathbb{k}k[z]→k (and k [ [ z ] ] k k [ [ z ] ] → k k[[z]]rarrk\mathbb{k} \llbracket z \rrbracket \rightarrow \mathbb{k}k[[z]]→k ) sending z 0 z ↦ 0 z|->0z \mapsto 0z↦0, we can regard every rational G G GGG-module V V VVV as a G [ z ] G [ z ] G[z]G[z]G[z]-module or a G [ [ z ] ] G [ [ z ] ] G[[z]]G \llbracket z \rrbracketG[[z]]-module with (trivial) G m rot G m rot  G_(m)^("rot ")\mathbb{G}_{m}^{\text {rot }}Gmrot -action through e v 0 e v 0 ev_(0)\mathrm{ev}_{0}ev0. We also have a G [ [ z ] ] G [ [ z ] ] G[[z]]G \llbracket z \rrbracketG[[z]]-module structure (without a G m rot G m rot  G_(m)^("rot ")\mathbb{G}_{m}^{\text {rot }}Gmrot -action) on V [ [ z ] ] := V k [ [ z ] ] V [ [ z ] ] := V ⊗ k [ [ z ] ] V[[z]]:=V oxk[[z]]V \llbracket z \rrbracket:=V \otimes \mathbb{k} \llbracket z \rrbracketV[[z]]:=V⊗k[[z]] that surjects onto V V VVV.
Definition 4.1 (global Weyl modules). Let ( λ ) ⨀ ( λ ) ⨀(lambda)\bigodot(\lambda)⨀(λ) be the category of rational G [ z ] G [ z ] G[z]G[z]G[z]-modules M M MMM that admits a decreasing filtration
M = F 0 M F 1 M F 2 M such that k 0 F k M = { 0 } M = F 0 M ⊃ F 1 M ⊃ F 2 M ⊃ ⋯  such that  ⋂ k ≥ 0   F k M = { 0 } M=F_(0)M supF_(1)M supF_(2)M sup cdotsquad" such that "nnn_(k >= 0)F_(k)M={0}M=F_{0} M \supset F_{1} M \supset F_{2} M \supset \cdots \quad \text { such that } \bigcap_{k \geq 0} F_{k} M=\{0\}M=F0M⊃F1M⊃F2M⊃⋯ such that ⋂k≥0FkM={0}
and each F k M / F k 1 M ( k 1 ) F k M / F k − 1 M ( k ≥ 1 ) F_(k)M//F_(k-1)M(k >= 1)F_{k} M / F_{k-1} M(k \geq 1)FkM/Fk−1M(k≥1) belongs to { q m L ( μ ) } m Z , λ μ X + q m L ( μ ) m ∈ Z , λ ≥ μ ∈ X + {q^(m)L(mu)}_(m inZ,lambda >= mu inX_(+))\left\{q^{m} L(\mu)\right\}_{m \in \mathbb{Z}, \lambda \geq \mu \in \mathbb{X}_{+}}{qmL(μ)}m∈Z,λ≥μ∈X+. For each λ X + λ ∈ X + lambda inX_(+)\lambda \in \mathbb{X}_{+}λ∈X+, we define the global Weyl module W ( λ ) W ( λ ) W(lambda)\mathbb{W}(\lambda)W(λ) of G [ z ] G [ z ] G[z]G[z]G[z] as the projective cover of L ( λ ) L ( λ ) L(lambda)L(\lambda)L(λ) in ( λ ) ⨀ ( λ ) ⨀(lambda)\bigodot(\lambda)⨀(λ).
Note that W ( λ ) W ( λ ) W(lambda)\mathbb{W}(\lambda)W(λ) automatically acquires a G m rot G m rot  G_(m)^("rot ")\mathbb{G}_{m}^{\text {rot }}Gmrot -action by its universality (as it exists).
Theorem 4.2. For each λ X + λ ∈ X + lambda inX_(+)\lambda \in \mathbb{X}_{+}λ∈X+with λ = i = 1 r m i ϖ i λ = ∑ i = 1 r   m i Ï– i lambda=sum_(i=1)^(r)m_(i)Ï–_(i)\lambda=\sum_{i=1}^{r} m_{i} \varpi_{i}λ=∑i=1rmiÏ–i, we have
End G [ z ] W ( λ ) i = 1 r k [ x i , 1 , , x i , m i ] m i End G [ z ] ⁡ W ( λ ) ≅ ⨂ i = 1 r   k x i , 1 , … , x i , m i â„‘ m i End_(G[z])W(lambda)~=⨂_(i=1)^(r)k[x_(i,1),dots,x_(i,m_(i))]^(â„‘_(m_(i)))\operatorname{End}_{G[z]} \mathbb{W}(\lambda) \cong \bigotimes_{i=1}^{r} \mathbb{k}\left[x_{i, 1}, \ldots, x_{i, m_{i}}\right]^{\Im_{m_{i}}}EndG[z]⁡W(λ)≅⨂i=1rk[xi,1,…,xi,mi]â„‘mi
where each x i , 1 , , x i , m i x i , 1 , … , x i , m i x_(i,1),dots,x_(i,m_(i))x_{i, 1}, \ldots, x_{i, m_{i}}xi,1,…,xi,mi is of degree one with respect to the G m r o t G m r o t G_(m)^(rot)\mathbb{G}_{m}^{\mathrm{rot}}Gmrot-action. In addition, the action of End G [ z ] W ( λ ) End G [ z ] ⁡ W ( λ ) End_(G[z])W(lambda)\operatorname{End}_{G[z]} \mathbb{W}(\lambda)EndG[z]⁡W(λ) on W ( λ ) W ( λ ) W(lambda)\mathbb{W}(\lambda)W(λ) is free.
Theorem 4.2 was proved by Fourier-Littelmann [25] (for k = C k = C k=C\mathbb{k}=\mathbb{C}k=C and G G GGG of type ADE), Naoi [72] (for k = C k = C k=C\mathbb{k}=\mathbb{C}k=C and G G GGG of type BCFG), and it was transferred to char k > 0 k > 0 k > 0\mathbb{k}>0k>0 in [50] using results from the global bases of quantum affine algebras [ 4 , 42 ] [ 4 , 42 ] [4,42][4,42][4,42].
By Theorem 4.2, we factor out the positive degree parts of End G [ z ] W ( λ ) End G [ z ] ⁡ W ( λ ) End_(G[z])W(lambda)\operatorname{End}_{G[z]} \mathbb{W}(\lambda)EndG[z]⁡W(λ) to obtain
We call it a local Weyl module of G [ z ] G [ z ] G[z]G[z]G[z].
The following result clarifies that our global/local Weyl modules are the best possible analogues of Weyl modules for G G GGG (see Theorem 2.1):
Theorem 4.3 (Chari-Ion [14] for char k = 0 k = 0 k=0\mathbb{k}=0k=0, and [50] + ε + ε +epsi+\varepsilon+ε for char k > 0 k > 0 k > 0\mathbb{k}>0k>0 ). For each λ , μ X + λ , μ ∈ X + lambda,mu inX_(+)\lambda, \mu \in \mathbb{X}_{+}λ,μ∈X+, we have
(4.1) Ext G [ z ] i ( W ( λ ) , W ( μ ) ) k δ i , 0 δ λ , μ (4.1) Ext G [ z ] i ⁡ W ( λ ) , W ( μ ) ∗ ≅ k ⊕ δ i , 0 δ λ , μ ∗ {:(4.1)Ext_(G[z])^(i)(W(lambda),W(mu)^(**))~=k^(o+delta_(i,0)delta_(lambda,mu^(**))):}\begin{equation*} \operatorname{Ext}_{G[z]}^{i}\left(\mathbb{W}(\lambda), W(\mu)^{*}\right) \cong \mathbb{k}^{\oplus \delta_{i, 0} \delta_{\lambda, \mu^{*}}} \tag{4.1} \end{equation*}(4.1)ExtG[z]i⁡(W(λ),W(μ)∗)≅k⊕δi,0δλ,μ∗
where μ μ ∗ mu^(**)\mu^{*}μ∗ is the highest weight of L ( μ ) L ( μ ) ∗ L(mu)^(**)L(\mu)^{*}L(μ)∗. By taking the graded Euler-Poincaré characteristic, (4.1) implies the orthogonality of Macdonald polynomials with respect to the Macdonald pairing specialized to t = 0 t = 0 t=0t=0t=0. In particular, gch W ( λ ) W ( λ ) W(lambda)W(\lambda)W(λ) and gch W ( λ ) W ( λ ) W(lambda)\mathbb{W}(\lambda)W(λ) do not depend on k k k\mathbb{k}k.
The proof of Theorem 4.3 in [ 50 , $ 3.3 ] [ 50 , $ 3.3 ] [50,$3.3][50, \$ 3.3][50,$3.3] relies on the adjoint property of the Demazure functors observed in [20, PROPOSITION 5.7] and systematically utilized in [15]. The case λ = μ λ = μ ∗ lambda=mu^(**)\lambda=\mu^{*}λ=μ∗ and i > 1 i > 1 i > 1i>1i>1 in Theorem 4.3 is not recorded in [50], and might appear elsewhere.

5. SEMI-INFINITE FLAG MANIFOLDS

We keep the setting of the previous section. In view of the projectivity of W ( λ ) W ( λ ) W(lambda)\mathbb{W}(\lambda)W(λ) 's in ( λ ) ⨀ ( λ ) ⨀(lambda)\bigodot(\lambda)⨀(λ) 's, we find unique degree-zero G [ z ] G [ z ] G[z]G[z]G[z]-module maps
(5.1) W ( λ + μ ) W ( λ ) W ( μ ) , λ X + (5.1) W ( λ + μ ) → W ( λ ) ⊗ W ( μ ) , λ ∈ X + {:(5.1)W(lambda+mu)rarrW(lambda)oxW(mu)","quad lambda inX_(+):}\begin{equation*} \mathbb{W}(\lambda+\mu) \rightarrow \mathbb{W}(\lambda) \otimes \mathbb{W}(\mu), \quad \lambda \in \mathbb{X}_{+} \tag{5.1} \end{equation*}(5.1)W(λ+μ)→W(λ)⊗W(μ),λ∈X+
Therefore, the recipe described in Section 2 equips
R G := λ X + W ( λ ) R G := ⨁ λ ∈ X +   W ( λ ) ∨ R_(G):=bigoplus_(lambda inX_(+))W(lambda)^(vv)R_{G}:=\bigoplus_{\lambda \in \mathbb{X}_{+}} \mathbb{W}(\lambda)^{\vee}RG:=⨁λ∈X+W(λ)∨
with a structure of a commutative algebra compatible with the action of G [ z ] G m rot × T G [ z ] ⋊ G m rot  × T G[z]><|G_(m)^("rot ")xx TG[z] \rtimes \mathbb{G}_{m}^{\text {rot }} \times TG[z]⋊Gmrot ×T. Since the G m rot G m rot  G_(m)^("rot ")\mathbb{G}_{m}^{\text {rot }}Gmrot -degree of R G R G R_(G)R_{G}RG is bounded from the above, the G [ z ] G [ z ] G[z]G[z]G[z]-action on R G R G R_(G)R_{G}RG automatically extends to the G [ [ z ] ] G [ [ z ] ] G[[z]]G \llbracket z \rrbracketG[[z]]-action. We set
Q G := ( Spec R G E ) / T Q G := Spec ⁡ R G ∖ E / T Q_(G):=(Spec R_(G)\\E)//T\mathbf{Q}_{G}:=\left(\operatorname{Spec} R_{G} \backslash E\right) / TQG:=(Spec⁡RG∖E)/T
where E E EEE is a closed subset of Spec R G R G R_(G)R_{G}RG on which the T T TTT-action is not free. Let us consider the G ( ( z ) ) G ( ( z ) ) G((z))G((z))G((z))-orbit of
(5.2) { [ v ϖ i ] } i = 1 r i = 1 r P ( V ( ϖ i ) ( ( z ) ) ) (5.2) v Ï– i i = 1 r ∈ ∏ i = 1 r   P V Ï– i ( ( z ) ) {:(5.2){[v_(Ï–_(i))]}_(i=1)^(r)inprod_(i=1)^(r)P(V(Ï–_(i))((z))):}\begin{equation*} \left\{\left[\mathbf{v}_{\varpi_{i}}\right]\right\}_{i=1}^{r} \in \prod_{i=1}^{r} \mathbb{P}\left(V\left(\varpi_{i}\right)((z))\right) \tag{5.2} \end{equation*}(5.2){[vÏ–i]}i=1r∈∏i=1rP(V(Ï–i)((z)))
viewed as a set of points, that we denote by Q G Q G Q_(G)\mathcal{Q}_{G}QG. By examining the coefficients of the defining relations of B B B\mathscr{B}B with its k ( ( z ) ) k ( ( z ) ) k((z))\mathbb{k}((z))k((z))-valued points, we find that the intersection
(5.3) Q G i = 1 r P ( V ( ϖ i ) [ [ z ] ] z m i ) i = 1 r P ( V ( ϖ i ) [ [ z ] ] z m i ) i = 1 r P ( V ( ϖ i ) ( ( z ) ) ) (5.3) Q G ∩ ∏ i = 1 r   P V Ï– i [ [ z ] ] z m i ⊂ ∏ i = 1 r   P V Ï– i [ [ z ] ] z m i ⊂ ∏ i = 1 r   P V Ï– i ( ( z ) ) {:(5.3)Q_(G)nnprod_(i=1)^(r)P(V(Ï–_(i))([[)z(]])z^(m_(i)))subprod_(i=1)^(r)P(V(Ï–_(i))([[)z(]])z^(m_(i)))subprod_(i=1)^(r)P(V(Ï–_(i))((z))):}\begin{equation*} \mathcal{Q}_{G} \cap \prod_{i=1}^{r} \mathbb{P}\left(V\left(\varpi_{i}\right) \llbracket z \rrbracket z^{m_{i}}\right) \subset \prod_{i=1}^{r} \mathbb{P}\left(V\left(\varpi_{i}\right) \llbracket z \rrbracket z^{m_{i}}\right) \subset \prod_{i=1}^{r} \mathbb{P}\left(V\left(\varpi_{i}\right)((z))\right) \tag{5.3} \end{equation*}(5.3)QG∩∏i=1rP(V(Ï–i)[[z]]zmi)⊂∏i=1rP(V(Ï–i)[[z]]zmi)⊂∏i=1rP(V(Ï–i)((z)))
defines a closed subscheme for any choice of m 1 , , m r Z m 1 , … , m r ∈ Z m_(1),dots,m_(r)inZm_{1}, \ldots, m_{r} \in \mathbb{Z}m1,…,mr∈Z. We denote this subscheme by Q G ( t β ) Q G t β Q_(G)(t_(beta))\mathbf{Q}_{G}\left(t_{\beta}\right)QG(tβ), where β = i = 1 r m i α i β = ∑ i = 1 r   m i α i ∨ beta=sum_(i=1)^(r)m_(i)alpha_(i)^(vv)\beta=\sum_{i=1}^{r} m_{i} \alpha_{i}^{\vee}β=∑i=1rmiαi∨ is an element of the dual lattice (coroot lattice) X X ∨ X^(vv)\mathbb{X}^{\vee}X∨ of X X X\mathbb{X}X equipped with a basis { α i } i = 1 r α i ∨ i = 1 r {alpha_(i)^(vv)}_(i=1)^(r)\left\{\alpha_{i}^{\vee}\right\}_{i=1}^{r}{αi∨}i=1r such that α i ( ϖ j ) = δ i , j α i ∨ Ï– j = δ i , j alpha_(i)^(vv)(Ï–_(j))=delta_(i,j)\alpha_{i}^{\vee}\left(\varpi_{j}\right)=\delta_{i, j}αi∨(Ï–j)=δi,j (i.e., α i α i ∨ alpha_(i)^(vv)\alpha_{i}^{\vee}αi∨ is a simple coroot). We note that P ( V ( ϖ i ) [ [ z ] ] z m i ) P V Ï– i [ [ z ] ] z m i P(V(Ï–_(i))([[)z(]])z^(m_(i)))\mathbb{P}\left(V\left(\varpi_{i}\right) \llbracket z \rrbracket z^{m_{i}}\right)P(V(Ï–i)[[z]]zmi) is a scheme, but it is not of finite type, and Q G ( t β ) Q G t β Q_(G)(t_(beta))\mathbf{Q}_{G}\left(t_{\beta}\right)QG(tβ) is also of infinite type.
Lemma 5.1. We have Q G ( t β ) Q G ( t γ ) Q G t β ≅ Q G t γ Q_(G)(t_(beta))~=Q_(G)(t_(gamma))\mathbf{Q}_{G}\left(t_{\beta}\right) \cong \mathbf{Q}_{G}\left(t_{\gamma}\right)QG(tβ)≅QG(tγ) for each pair β , γ X β , γ ∈ X ∨ beta,gamma inX^(vv)\beta, \gamma \in \mathbb{X}^{\vee}β,γ∈X∨ as schemes equipped with G [ [ z ] ] G [ [ z ] ] G[[z]]G \llbracket z \rrbracketG[[z]]-actions. Hence, the union
Q G rat = β Q G ( t β ) Q G rat  = ⋃ β   Q G t β Q_(G)^("rat ")=uuu_(beta)Q_(G)(t_(beta))\mathbf{Q}_{G}^{\text {rat }}=\bigcup_{\beta} \mathbf{Q}_{G}\left(t_{\beta}\right)QGrat =⋃βQG(tβ)
is a pure ind-scheme of ind-infinite type equipped with the action of G [ [ z ] ] G m r o t G [ [ z ] ] ⋊ G m r o t G[[z]]><|G_(m)^(rot)G \llbracket z \rrbracket \rtimes \mathbb{G}_{m}^{\mathrm{rot}}G[[z]]⋊Gmrot. Moreover, the set of G [ [ z ] ] G [ [ z ] ] G[[z]]G \llbracket z \rrbracketG[[z]]-orbits in Q G rat Q G rat  Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat  is in bijection with X X ∨ X^(vv)\mathbb{X}^{\vee}X∨.
In effect, we have an open dense G [ [ z ] ] G [ [ z ] ] G[[z]]G \llbracket z \rrbracketG[[z]]-orbit O G ( t β ) Q G ( t β ) O G t β ⊂ Q G t β O_(G)(t_(beta))subQ_(G)(t_(beta))\mathbf{O}_{G}\left(t_{\beta}\right) \subset \mathbf{Q}_{G}\left(t_{\beta}\right)OG(tβ)⊂QG(tβ) that is isomorphic to G [ [ z ] ] / ( T N [ [ z ] ] ) G [ [ z ] ] / ( T â‹… N [ [ z ] ] ) G[[z]]//(T*N[[z]])G \llbracket z \rrbracket /(T \cdot N \llbracket z \rrbracket)G[[z]]/(Tâ‹…N[[z]]). By the Bruhat decomposition, we divide O G ( t β ) O G t β O_(G)(t_(beta))\mathbf{O}_{G}\left(t_{\beta}\right)OG(tβ) into the disjoint union of I I I\mathbf{I}I-orbits as w W O ( w t β ) ⨆ w ∈ W   O w t β ⨆_(w in W)O(wt_(beta))\bigsqcup_{w \in W} \mathbf{O}\left(w t_{\beta}\right)⨆w∈WO(wtβ) such that O ( t β ) O G ( t β ) O t β ⊂ O G t β O(t_(beta))subO_(G)(t_(beta))\mathbf{O}\left(t_{\beta}\right) \subset \mathbf{O}_{G}\left(t_{\beta}\right)O(tβ)⊂OG(tβ) is open dense. Identifying β X β ∈ X ∨ beta inXvv\beta \in \mathbb{X} \veeβ∈X∨ with t β t β t_(beta)t_{\beta}tβ, we set W a f := W X W a f := W ⋉ X ∨ W_(af):=W|><X^(vv)W_{\mathrm{af}}:=W \ltimes \mathbb{X}^{\vee}Waf:=W⋉X∨. We define
Q G ( w ) := O ( w ) ¯ Q G r a t , w W a f Q G ( w ) := O ( w ) ¯ ⊂ Q G r a t , w ∈ W a f Q_(G)(w):= bar(O(w))subQ_(G)^(rat),quad w inW_(af)\mathbf{Q}_{G}(w):=\overline{\mathbf{O}(w)} \subset \mathbf{Q}_{G}^{\mathrm{rat}}, \quad w \in W_{\mathrm{af}}QG(w):=O(w)¯⊂QGrat,w∈Waf
The inclusion relation on { Q G ( w ) } w W a f Q G ( w ) w ∈ W a f {Q_(G)(w)}_(w inW_(af))\left\{\mathbf{Q}_{G}(w)\right\}_{w \in W_{\mathrm{af}}}{QG(w)}w∈Waf is described by the generic Bruhat order [62]. We refer to the partial order on W af W af  W_("af ")W_{\text {af }}Waf  induced from this closure ordering by 2 ≤ ∞ 2 <= (oo)/(2)\leq \frac{\infty}{2}≤∞2 as in [ 50 , 52 ] [ 50 , 52 ] [50,52][50,52][50,52] (there we sometimes called 2 ≤ ∞ 2 <= (oo)/(2)\leq \frac{\infty}{2}≤∞2 as the semi-infinite Bruhat order).
Theorem 5.2. The scheme Q G ( w ) Q G ( w ) Q_(G)(w)\mathbf{Q}_{G}(w)QG(w) is normal for each w W a f w ∈ W a f w inW_(af)w \in W_{\mathrm{af}}w∈Waf. In addition, the ind-scheme Q G rat Q G rat  Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat  is a strict ind-scheme in the sense that each inclusion is a closed immersion. The indscheme Q G rat Q G rat  Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat  coarsely ind-represents the coset G ( ( z ) ) / ( T N ( ( z ) ) ) G ( ( z ) ) / ( T ⋅ N ( ( z ) ) ) G((z))//(T*N((z)))G((z)) /(T \cdot N((z)))G((z))/(T⋅N((z))).
The first two statements are proved in [52] when char k = 0 k = 0 k=0k=0k=0. The proof valid for char k 2 k ≠ 2 k!=2\mathbb{k} \neq 2k≠2, as well as the last assertion, are contained in [50]. This last assertion says that the (ind-)scheme Q G rat Q G rat  Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat  is the universal one that maps to every (ind-)scheme whose points yield Q G Q G Q_(G)Q_{G}QG. It follows that if we take a family { Y ( λ ) } λ X + { Y ( λ ) } λ ∈ X + {Y(lambda)}_(lambda inX_(+))\{\mathbb{Y}(\lambda)\}_{\lambda \in \mathbb{X}_{+}}{Y(λ)}λ∈X+instead of { W ( λ ) } λ X + { W ( λ ) } λ ∈ X + {W(lambda)}_(lambda inX_(+))\{\mathbb{W}(\lambda)\}_{\lambda \in \mathbb{X}_{+}}{W(λ)}λ∈X+to define Q G ( t β ) Q G t β Q_(G)(t_(beta))\mathbf{Q}_{G}\left(t_{\beta}\right)QG(tβ), then the corresponding coordinate ring R G R G ′ R_(G)^(')R_{G}^{\prime}RG′ admits a map to R G R G R_(G)R_{G}RG. Let us point out that this can be thought of as a family version of the properties of global Weyl modules discussed in Section 4, and we indeed have several reasonable choices of { Y ( λ ) } λ X + { Y ( λ ) } λ ∈ X + {Y(lambda)}_(lambda inX_(+))\{\mathbb{Y}(\lambda)\}_{\lambda \in \mathbb{X}_{+}}{Y(λ)}λ∈X+other than { W ( λ ) } λ X + { W ( λ ) } λ ∈ X + {W(lambda)}_(lambda inX_(+))\{\mathbb{W}(\lambda)\}_{\lambda \in \mathbb{X}_{+}}{W(λ)}λ∈X+including the coordinate ring of the arc scheme of G / N G / N G//NG / NG/N. For simplicity, we may refer to Q G ( t 0 ) Q G t 0 Q_(G)(t_(0))\mathbf{Q}_{G}\left(t_{0}\right)QG(t0) as Q G Q G Q_(G)\mathbf{Q}_{G}QG below.
The inclusion
(5.4) Q G i = 1 r P ( V ( ϖ i ) [ [ z ] ] ) (5.4) Q G ⊂ ∏ i = 1 r   P V Ï– i [ [ z ] ] {:(5.4)Q_(G)subprod_(i=1)^(r)P(V(Ï–_(i))([[)z(]])):}\begin{equation*} \mathbf{Q}_{G} \subset \prod_{i=1}^{r} \mathbb{P}\left(V\left(\varpi_{i}\right) \llbracket z \rrbracket\right) \tag{5.4} \end{equation*}(5.4)QG⊂∏i=1rP(V(Ï–i)[[z]])
induces a line bundle O Q G ( ϖ i ) O Q G Ï– i O_(Q_(G))(Ï–_(i))\mathcal{O}_{\mathbf{Q}_{G}}\left(\varpi_{i}\right)OQG(Ï–i) on Q G Q G Q_(G)\mathbf{Q}_{G}QG, that is, the pull-back of O ( 1 ) O ( 1 ) O(1)\mathcal{O}(1)O(1) from P ( V ( ϖ i ) [ [ z ] ] ) P V Ï– i [ [ z ] ] P(V(Ï–_(i))([[)z(]]))\mathbb{P}\left(V\left(\varpi_{i}\right) \llbracket z \rrbracket\right)P(V(Ï–i)[[z]]). By taking the tensor products, we have O Q G ( λ ) := i = 1 r O Q G ( ϖ i ) n i O Q G ( λ ) := ⨂ i = 1 r   O Q G Ï– i ⊗ n i O_(Q_(G))(lambda):=⨂_(i=1)^(r)O_(Q_(G))(Ï–_(i))^(oxn_(i))\mathcal{O}_{\mathbf{Q}_{G}}(\lambda):=\bigotimes_{i=1}^{r} \mathcal{O}_{\mathbf{Q}_{G}}\left(\varpi_{i}\right)^{\otimes n_{i}}OQG(λ):=⨂i=1rOQG(Ï–i)⊗ni for λ = i = 1 r n i ϖ i λ = ∑ i = 1 r   n i Ï– i lambda=sum_(i=1)^(r)n_(i)Ï–_(i)\lambda=\sum_{i=1}^{r} n_{i} \varpi_{i}λ=∑i=1rniÏ–i ( n i Z ) n i ∈ Z (n_(i)inZ)\left(n_{i} \in \mathbb{Z}\right)(ni∈Z). By Lemma 5.1, we have O Q G rat ( λ ) ( λ X ) O Q G rat  ( λ ) ( λ ∈ X ) O_(Q_(G)^("rat "))(lambda)(lambda inX)\mathcal{O}_{\mathbf{Q}_{G}^{\text {rat }}}(\lambda)(\lambda \in \mathbb{X})OQGrat (λ)(λ∈X) on Q G rat Q G rat  Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat  that yields O Q G ( λ ) O Q G ( λ ) O_(Q_(G))(lambda)\mathcal{O}_{\mathbf{Q}_{G}}(\lambda)OQG(λ) by restriction.
Theorem 5.3 ([52] for char k = 0 k = 0 k=0\mathbb{k}=0k=0, and [50] for char k 2 k ≠ 2 k!=2\mathbb{k} \neq 2k≠2 ). For each λ X λ ∈ X lambda inX\lambda \in \mathbb{X}λ∈X, we have
H i ( Q G , O Q G ( λ ) ) { W ( λ ) , i = 0 , λ X + { 0 } , otherwise H i Q G , O Q G ( λ ) ∨ ≅ W ( λ ) ,      i = 0 , λ ∈ X + { 0 } ,       otherwise  H^(i)(Q_(G),O_(Q_(G))(lambda))^(vv)~={[W(lambda)",",i=0","lambda inX_(+)],[{0}","," otherwise "]:}H^{i}\left(\mathbf{Q}_{G}, \mathcal{O}_{\mathbf{Q}_{G}}(\lambda)\right)^{\vee} \cong \begin{cases}\mathbb{W}(\lambda), & i=0, \lambda \in \mathbb{X}_{+} \\ \{0\}, & \text { otherwise }\end{cases}Hi(QG,OQG(λ))∨≅{W(λ),i=0,λ∈X+{0}, otherwise 
The proof of Theorem 5.3 depends on the freeness of R G R G R_(G)R_{G}RG over an infinitely-manyvariable polynomial ring, that yields a regular sequence of infinite length. Such a situation never occur for finite type schemes, or infinite type schemes like B C thick B C thick  B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }}BCthick . In case G = SL ( 2 ) G = SL ⁡ ( 2 ) G=SL(2)G=\operatorname{SL}(2)G=SL⁡(2), Theorem 5.3 reduces to an exercise in algebraic geometry by Q G P ( k 2 [ [ z ] ] ) Q G ≅ P k 2 [ [ z ] ] Q_(G)~=P(k^(2)([[)z(]]))\mathbf{Q}_{G} \cong \mathbb{P}\left(\mathbb{k}^{2} \llbracket z \rrbracket\right)QG≅P(k2[[z]]).
Theorem 5.3 has an ind-model counterpart proved earlier [10]. The Frobenius splitting of Q G Q G Q_(G)\mathbf{Q}_{G}QG (explained later) and Theorem 5.3 imply this ind-model counterpart. However, the author is uncertain whether [10] implies Theorem 5.3 (even in case char k = 0 k = 0 k=0k=0k=0 ) since the natural ring coming from the ind-model is a completion of R G R G R_(G)R_{G}RG, and the completion operation of a ring loses information in general. We have an analogue of Theorem 5.3 for all I I I\mathbf{I}I-orbit closures, proved for the ind-model in [46,50] and for the formal model in [50,52].

6. FROBENIUS SPLITTINGS

We continue to work in the setting of the previous section. We fix a prime p > 0 p > 0 p > 0p>0p>0. For a scheme X X X\mathfrak{X}X over F p F p F_(p)\mathbb{F}_{p}Fp, we have a Frobenius morphism F r : X X F r : X → X Fr:XrarrX\mathrm{Fr}: \mathfrak{X} \rightarrow \mathfrak{X}Fr:X→X induced from the p p ppp th power map. We have a natural map Fr O X O X Fr ∗ ⁡ O X → O X Fr^(**)O_(X)rarrO_(X)\operatorname{Fr}^{*} \mathcal{O}_{\mathfrak{X}} \rightarrow \mathcal{O}_{\mathfrak{X}}Fr∗⁡OX→OX that induces a map O X F r O X O X → F r ∗ O X O_(X)rarrFr_(**)O_(X)\mathcal{O}_{\mathfrak{X}} \rightarrow \mathrm{Fr}_{*} \mathcal{O}_{\mathfrak{X}}OX→Fr∗OX by adjunction. The Frobenius splitting ϕ : Fr O X O X Ï• : Fr ∗ ⁡ O X → O X phi:Fr_(**)O_(X)rarrO_(X)\phi: \operatorname{Fr}_{*} \mathcal{O}_{\mathfrak{X}} \rightarrow \mathcal{O}_{\mathfrak{X}}Ï•:Fr∗⁡OX→OX is an O X -module map such that the composition O X -module map such that the composition  O_(X"-module map such that the composition ")\mathcal{O}_{\mathfrak{X} \text {-module map such that the composition }}OX-module map such that the composition 
O X F r O X ϕ O X O X → F r ∗ O X → Ï• O X O_(X)rarrFr_(**)O_(X)rarr"phi"O_(X)\mathcal{O}_{\mathfrak{X}} \rightarrow \mathrm{Fr}_{*} \mathcal{O}_{\mathfrak{X}} \xrightarrow{\phi} \mathcal{O}_{\mathfrak{X}}OX→Fr∗OX→ϕOX
is the identity. If X X X\mathfrak{X}X is projective (and is of finite type) and O X O X O_(X)\mathcal{O}_{\mathfrak{X}}OX admits a Frobenius splitting, then X X X\mathfrak{X}X is reduced and an ample line bundle has the higher cohomology vanishing [68].
For generality on Frobenius splittings, as well as their applications to B B B\mathscr{B}B and B C thin B C thin  B_(C)^("thin ")\mathscr{B}_{C}^{\text {thin }}BCthin , we refer to Brion-Kumar [12] (note that [12] has a finite type assumption, that we drop in case the proof does not require it. In the paragraph above, reducedness does not require the finite type assumption, while the higher cohomology vanishing requires the finite type assumption through the Serre vanishing). Frobenius splitting of B B B\mathscr{B}B in char k = p k = p k=p\mathbb{k}=pk=p is useful in proving that Schubert and Richardson varieties are reduced, normal, and have rational singularities. There are two major ways to construct a Frobenius splitting of B B B\mathscr{B}B : one is to investigate the global section of the ( 1 p ) ( 1 − p ) (1-p)(1-p)(1−p) th power of the canonical bundle, and the other is to use a Bott-Samelson-Demazure-Hansen (=BSDH) resolution of B B B\mathscr{B}B.
Since B C thin B C thin  B_(C)^("thin ")\mathscr{B}_{C}^{\text {thin }}BCthin  is no longer smooth, we cannot use the canonical bundle to construct a Frobenius splitting. Nevertheless, a (partial) BSDH resolution does the job. The situation of B C thick B C thick  B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }}BCthick  is a bit worse. The canonical bundle of B C thick B C thick  B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }}BCthick  makes some sense, but the author does not know whether it has enough power to produce a Frobenius splitting. The scheme B C thick B C thick  B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }}BCthick 
admits a BSDH resolution, but it is a successive P 1 P 1 P^(1)\mathbb{P}^{1}P1-fibration over an infinite-type scheme. Thus, we cannot equip B C thick B C thick  B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }}BCthick  with a Frobenius splitting by either of the above means at present. Despite this, we can transfer a Frobenius splitting of B C thin B C thin  B_(C)^("thin ")\mathscr{B}_{C}^{\text {thin }}BCthin  to B C thick B C thick  B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }}BCthick  by using the compatible splitting property of a point [49], following an idea of Mathieu.
Frobenius splitting of Q G rat Q G rat  Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat  (or rather each of its ind-piece Q G ( w ) Q G ( w ) Q_(G)(w)\mathbf{Q}_{G}(w)QG(w) ) is used below, and hence we need a recipe to produce one. However, the situation of the BSDH resolution is similar to that of B C thick B C thick  B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }}BCthick , and the canonical bundle on Q G rat Q G rat  Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat  simply does not make sense naively (e.g., its T T TTT-weight at a point must be infinity). Therefore, we need a new proof strategy. Our strategy in [50] is to regard R G R G R_(G)R_{G}RG as a subalgebra of the corresponding coordinate ring of B C thick B C thick  B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }}BCthick , and prove that a Frobenius splitting of B C thick B C thick  B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }}BCthick  preserves R G R G R_(G)R_{G}RG. For this, we first see that each W ( m λ ) ( m Z > 0 , λ X + ) W ( m λ ) m ∈ Z > 0 , λ ∈ X + W(m lambda)(m inZ_( > 0),lambda inX_(+))\mathbb{W}(m \lambda)\left(m \in \mathbb{Z}_{>0}, \lambda \in \mathbb{X}_{+}\right)W(mλ)(m∈Z>0,λ∈X+)is a quotient of L ( m Λ ) L ( m Λ ) L(m Lambda)L(m \Lambda)L(mΛ) for some Λ P + Λ ∈ P + Lambda inP_(+)\Lambda \in P_{+}Λ∈P+by twisting the G [ z 1 ] G z − 1 G[z^(-1)]G\left[z^{-1}\right]G[z−1]-action into a G [ z ] G [ z ] G[z]G[z]G[z]-action as z 1 z z − 1 ↦ z z^(-1)|->zz^{-1} \mapsto zz−1↦z. Let π m : L ( m Λ ) W ( m λ ) Ï€ m : L ( m Λ ) → W ( m λ ) pi_(m):L(m Lambda)rarrW(m lambda)\pi_{m}: L(m \Lambda) \rightarrow \mathbb{W}(m \lambda)Ï€m:L(mΛ)→W(mλ) be the quotient map. This embeds (a suitable Z Z Z\mathbb{Z}Z-graded subalgebra of) R G R G R_(G)R_{G}RG into (3.1) as an algebra with G [ [ z ] ] G m rot G [ [ z ] ] ⋉ G m rot  G[[z]]|><G_(m)^("rot ")G \llbracket z \rrbracket \ltimes \mathbb{G}_{m}^{\text {rot }}G[[z]]⋉Gmrot -action. We need to show that the map ϕ Ï• ∨ phi^(vv)\phi^{\vee}ϕ∨ obtained by dualizing the Frobenius splitting of B C thick B C thick  B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }}BCthick  induces a map ϕ W Ï• W ∨ phi_(W)^(vv)\phi_{\mathbb{W}}^{\vee}Ï•W∨ in the following diagram:
This is equivalent to seeing that ϕ ( ker π m ) ker π p m Ï• ∨ ker ⁡ Ï€ m ⊂ ker ⁡ Ï€ p m phi^(vv)(ker pi_(m))sub ker pi_(pm)\phi^{\vee}\left(\operatorname{ker} \pi_{m}\right) \subset \operatorname{ker} \pi_{p m}ϕ∨(ker⁡πm)⊂ker⁡πpm. We use the projectivity of W ( m λ ) W ( m λ ) W(m lambda)\mathbb{W}(m \lambda)W(mλ) in ( m λ ) ⨀ ( m λ ) ⨀(m lambda)\bigodot(m \lambda)⨀(mλ) to assume that the G [ z ] G [ z ] G[z]G[z]G[z]-module generators of ker π m Ï€ m pi_(m)\pi_{m}Ï€m have T T TTT-weights that do not appear in W ( m λ ) W ( m λ ) W(m lambda)\mathbb{W}(m \lambda)W(mλ). In view of the fact that ker π p m Ï€ p m pi_(pm)\pi_{p m}Ï€pm contains all the T T TTT-weight spaces in L ( p m Λ ) L ( p m Λ ) L(pm Lambda)L(p m \Lambda)L(pmΛ) whose T T TTT-weights do not appear in W ( p m λ ) W ( p m λ ) W(pm lambda)\mathbb{W}(p m \lambda)W(pmλ), we have necessarily ϕ ( ker π m ) Ï• ∨ ker ⁡ Ï€ m ⊂ phi^(vv)(ker pi_(m))sub\phi^{\vee}\left(\operatorname{ker} \pi_{m}\right) \subsetϕ∨(ker⁡πm)⊂ ker π p m ker ⁡ Ï€ p m ker pi_(pm)\operatorname{ker} \pi_{p m}ker⁡πpm by the T T TTT-weight comparison of the generators.
In fact, every L ( Λ ) L ( Λ ) L(Lambda)L(\Lambda)L(Λ) admits a filtration by global Weyl modules when char k = 0 k = 0 k=0\mathbb{k}=0k=0 if we twist the action of G [ z ] G [ z ] G[z]G[z]G[z] on global Weyl modules into G [ z 1 ] [ 51 ] G z − 1 [ 51 ] G[z^(-1)][51]G\left[z^{-1}\right][51]G[z−1][51]. Therefore, we indeed obtain a Frobenius splitting of Q G Q G Q_(G)\mathbf{Q}_{G}QG via a novel proof based on the "universality" of the global Weyl module W ( λ ) W ( λ ) W(lambda)\mathbb{W}(\lambda)W(λ) explained in Section 4. In conclusion, we have:
Theorem 6.1 ([50, THeORem B]). The ind-scheme Q G rat Q G rat  Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat  admits a Frobenius splitting that is compatible with all I I I\mathbf{I}I-orbits when char k > 2 k > 2 k > 2\mathbb{k}>2k>2.

7. CONNECTION TO THE SPACE OF RATIONAL MAPS

Keep the setting as in Section 5. Let us consider the vector space embedding k ( ( z ) ) k [ [ z , z 1 ] ] k ( ( z ) ) ⊂ k [ [ z , z − 1 ] ] k((z))subk[[z,z^(-1)]]\mathbb{k}((z)) \subset \mathbb{k} \llbracket z, z^{-1} \rrbracketk((z))⊂k[[z,z−1]] into the formal power series with unbounded powers. The space k [ [ z , z 1 ] ] k [ [ z , z − 1 ] ] k[[z,z^(-1)]]\mathbb{k} \llbracket z, z^{-1} \rrbracketk[[z,z−1]] no longer forms a ring. Nevertheless, we have an automorphism of k [ [ z , z 1 ] ] k [ [ z , z − 1 ] ] k[[z,z^(-1)]]\mathbb{k} \llbracket z, z^{-1} \rrbracketk[[z,z−1]] by swapping z z zzz with z 1 z − 1 z^(-1)z^{-1}z−1. Together with the Chevalley involution of G G GGG (an automorphism of G G GGG that sends each element of T T TTT to its inverse), it induces an involution θ θ theta\thetaθ on the ambient space
Q G r a t i = 1 r P ( V ( ϖ i ) [ [ z , z 1 ] ] ) Q G r a t ⊂ ∏ i = 1 r   P V Ï– i [ [ z , z − 1 ] ] Q_(G)^(rat)subprod_(i=1)^(r)P(V(Ï–_(i))([[)z,z^(-1)(]]))\mathbf{Q}_{G}^{\mathrm{rat}} \subset \prod_{i=1}^{r} \mathbb{P}\left(V\left(\varpi_{i}\right) \llbracket z, z^{-1} \rrbracket\right)QGrat⊂∏i=1rP(V(Ï–i)[[z,z−1]])
We remark that θ θ theta\thetaθ induces an automorphism of G G GGG such that B θ ( B ) = T B ∩ θ ( B ) = T B nn theta(B)=TB \cap \theta(B)=TB∩θ(B)=T. Let w 0 w 0 w_(0)w_{0}w0 be the longest element in W W WWW.
Theorem 7.1 ([50, THEOREM B]). For all w , v W a f w , v ∈ W a f w,v inW_(af)w, v \in W_{\mathrm{af}}w,v∈Waf, the scheme-theoretic intersection Q G ( w ) θ ( Q G ( v w 0 ) ) Q G ( w ) ∩ θ Q G v w 0 Q_(G)(w)nn theta(Q_(G)(vw_(0)))\mathbf{Q}_{G}(w) \cap \theta\left(\mathbf{Q}_{G}\left(v w_{0}\right)\right)QG(w)∩θ(QG(vw0)) is reduced (we denote this intersection by Q G ( w , v ) Q G ( w , v ) Q_(G)(w,v)\mathcal{Q}_{G}(w, v)QG(w,v) and call it a Richardson variety of Q G rat Q G rat  Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat  below). It is normal when char k = 0 k = 0 k=0\mathbb{k}=0k=0 or char k 0 k ≫ 0 k≫0\mathbb{k} \gg 0k≫0.
The scheme Q G ( w , v ) Q G ( w , v ) Q_(G)(w,v)\mathcal{Q}_{G}(w, v)QG(w,v) is always of finite type, and the case w , v W w , v ∈ W w,v in Ww, v \in Ww,v∈W yields a Richardson variety of B B B\mathscr{B}B. The normality part of the proof of Theorem 7.1 goes as follows: Our Frobenius splitting of Q G rat Q G rat  Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat  induces a Frobenius splitting of Q G ( w , v ) Q G ( w , v ) Q_(G)(w,v)\mathcal{Q}_{G}(w, v)QG(w,v). In particular, it is reduced and weakly normal in char k > 2 k > 2 k > 2\mathbb{k}>2k>2. (Here a weakly normal ring is essentially a normal ring up to topology.) Then, we lift the weak normality to characteristic zero and prove the normality of the intersection by a geometric consideration. Once we deduce the normality in characteristic zero, we can reduce it to char k 0 k ≫ 0 k≫0\mathbb{k} \gg 0k≫0 by a general result.
Let us exhibit some relevant geometric considerations here. To this end, we assume k = C k = C k=C\mathbb{k}=\mathbb{C}k=C in the rest of this section. Recall that H 2 ( B , Z ) X H 2 ( B , Z ) ≅ X ∨ H_(2)(B,Z)~=X^(vv)H_{2}(\mathscr{B}, \mathbb{Z}) \cong \mathbb{X}^{\vee}H2(B,Z)≅X∨. Let E B 2 , β E B 2 , β EB_(2,beta)\mathcal{E} \mathscr{B}_{2, \beta}EB2,β (resp. B 2 , β ) B 2 , β {:B_(2,beta))\left.\mathscr{B}_{2, \beta}\right)B2,β) be the space of genus-zero stable maps with two marked points to ( P 1 × B ) ( P 1 × B ( (P^(1)xxB)(\left(\mathbb{P}^{1} \times \mathscr{B}\right)((P1×B)( resp. B ) B ) B)\mathscr{B})B) whose image has class ( 1 , β ) H 2 ( P 1 × B , Z ) ( 1 , β ) ∈ H 2 P 1 × B , Z (1,beta)inH_(2)(P^(1)xxB,Z)(1, \beta) \in H_{2}\left(\mathbb{P}^{1} \times \mathscr{B}, \mathbb{Z}\right)(1,β)∈H2(P1×B,Z) (resp. β H 2 ( B , Z ) β ∈ H 2 ( B , Z ) beta inH_(2)(B,Z)\beta \in H_{2}(\mathscr{B}, \mathbb{Z})β∈H2(B,Z) ), regarded as an algebraic variety with rational singularities [28]. We have a subvariety E B 2 , β b E B 2 , β b EB_(2,beta)^(b)\mathscr{E} \mathscr{B}_{2, \beta}^{b}EB2,βb such that the first marked point lands in 0 P 1 0 ∈ P 1 0inP^(1)0 \in \mathbb{P}^{1}0∈P1 and the second marked point lands in P 1 ∞ ∈ P 1 oo inP^(1)\infty \in \mathbb{P}^{1}∞∈P1 through the composition
( C , { x 1 , x 2 } ) f P 1 × B p r 1 P 1 C , x 1 , x 2 → f P 1 × B → p r 1 P 1 (C,{x_(1),x_(2)})rarr"f"P^(1)xxBrarr"pr_(1)"P^(1)\left(C,\left\{x_{1}, x_{2}\right\}\right) \xrightarrow{f} \mathbb{P}^{1} \times \mathscr{B} \xrightarrow{\mathrm{pr}_{1}} \mathbb{P}^{1}(C,{x1,x2})→fP1×B→pr1P1
Consider the Schubert variety (a B B BBB-orbit closure) B ( w ) B B ( w ) ⊂ B B(w)subB\mathscr{B}(w) \subset \mathscr{B}B(w)⊂B corresponding to w W w ∈ W w in Ww \in Ww∈W and the opposite Schubert variety (a θ ( B ) θ ( B ) theta(B)\theta(B)θ(B)-orbit closure) B o p ( v ) B B o p ( v ) ⊂ B B^(op)(v)subB\mathscr{B}^{\mathrm{op}}(v) \subset \mathscr{B}Bop(v)⊂B corresponding to v W v ∈ W v in Wv \in Wv∈W.

define
E B β ( w , v ) := ev 1 1 ( B ( w ) ) ev 2 1 ( B o p ( v ) ) E B β ( w , v ) := ev 1 − 1 ⁡ ( B ( w ) ) ∩ ev 2 − 1 ⁡ B o p ( v ) EB_(beta)(w,v):=ev_(1)^(-1)(B(w))nnev_(2)^(-1)(B^(op)(v))\mathscr{E} \mathscr{B}_{\beta}(w, v):=\operatorname{ev}_{1}^{-1}(\mathscr{B}(w)) \cap \operatorname{ev}_{2}^{-1}\left(\mathscr{B}^{\mathrm{op}}(v)\right)EBβ(w,v):=ev1−1⁡(B(w))∩ev2−1⁡(Bop(v))
Similarly, let e i : B 2 , β B ( i = 1 , 2 ) e i : B 2 , β → B ( i = 1 , 2 ) e_(i):B_(2,beta)rarrB(i=1,2)\mathrm{e}_{i}: \mathscr{B}_{2, \beta} \rightarrow \mathscr{B}(i=1,2)ei:B2,β→B(i=1,2) be the evaluation maps. For all w , v W w , v ∈ W w,v in Ww, v \in Ww,v∈W and β X β ∈ X ∨ beta inX^(vv)\beta \in \mathbb{X}^{\vee}β∈X∨, we set B β ( w , v ) := ( e 1 1 ( B ( w ) ) e 2 1 ( B op ( v ) ) ) B β ( w , v ) := e 1 − 1 ( B ( w ) ) ∩ e 2 − 1 B op  ( v ) B_(beta)(w,v):=(e_(1)^(-1)(B(w))nne_(2)^(-1)(B^("op ")(v)))\mathscr{B}_{\beta}(w, v):=\left(e_{1}^{-1}(\mathscr{B}(w)) \cap e_{2}^{-1}\left(\mathscr{B}^{\text {op }}(v)\right)\right)Bβ(w,v):=(e1−1(B(w))∩e2−1(Bop (v))). Let Q G ( β ) Q ∘ G ( β ) Q^(@)_(G)(beta)\stackrel{\circ}{Q}_{G}(\beta)Q∘G(β) denote the space of maps from P 1 P 1 P^(1)\mathbb{P}^{1}P1 to B B B\mathscr{B}B of degree β β beta\betaβ. By adding the identity map to P 1 P 1 P^(1)\mathbb{P}^{1}P1, each point of Q G ( β ) Q ∘ G ( β ) Q^(@)_(G)(beta)\stackrel{\circ}{Q}_{G}(\beta)Q∘G(β) yields a map P 1 ( P 1 × B ) P 1 → P 1 × B P^(1)rarr(P^(1)xxB)\mathbb{P}^{1} \rightarrow\left(\mathbb{P}^{1} \times \mathscr{B}\right)P1→(P1×B) of degree ( 1 , β ) ( 1 , β ) (1,beta)(1, \beta)(1,β). In addition, the identification of two P 1 P 1 P^(1)\mathbb{P}^{1}P1 's completely determines the marked points. Hence we have an inclusion Q G ( β ) E B 2 , β b Q ∘ G ( β ) ⊂ E B 2 , β b Q^(@)_(G)(beta)subEB_(2,beta)^(b)\stackrel{\circ}{Q}_{G}(\beta) \subset \mathscr{E} \mathscr{B}_{2, \beta}^{b}Q∘G(β)⊂EB2,βb.
Let Q G ( β ) ( β X ) Q G ( β ) β ∈ X ∨ Q_(G)(beta)(beta inX^(vv))\mathcal{Q}_{G}(\beta)\left(\beta \in \mathbb{X}^{\vee}\right)QG(β)(β∈X∨) denote the space of quasimaps from P 1 P 1 P^(1)\mathbb{P}^{1}P1 to B B B\mathscr{B}B of degree β β beta\betaβ [22], that is, a natural compactification of Q G ( β ) Q ∘ G ( β ) Q^(@)_(G)(beta)\stackrel{\circ}{Q}_{G}(\beta)Q∘G(β) such that
Q G ( β ) = 0 γ β Q G ( β γ ) × ( P 1 ) γ Q G ( β ) = ⨆ 0 ≤ γ ≤ β   Q ∘ G ( β − γ ) × P 1 γ Q_(G)(beta)=⨆_(0 <= gamma <= beta)Q^(@)_(G)(beta-gamma)xx(P^(1))^(gamma)\mathcal{Q}_{G}(\beta)=\bigsqcup_{0 \leq \gamma \leq \beta} \stackrel{\circ}{Q}_{G}(\beta-\gamma) \times\left(\mathbb{P}^{1}\right)^{\gamma}QG(β)=⨆0≤γ≤βQ∘G(β−γ)×(P1)γ
where γ β γ ≤ β gamma <= beta\gamma \leq \betaγ≤β is defined as β γ i = 1 r Z 0 α i β − γ ∈ ∑ i = 1 r   Z ≥ 0 α i ∨ beta-gamma insum_(i=1)^(r)Z_( >= 0)alpha_(i)^(vv)\beta-\gamma \in \sum_{i=1}^{r} \mathbb{Z}_{\geq 0} \alpha_{i}^{\vee}β−γ∈∑i=1rZ≥0αi∨, and
( P 1 ) γ = i = 1 r ( ( P 1 ) m i / S m i ) where γ = i = 1 r m i α i P 1 γ = ∏ i = 1 r   P 1 m i / S m i  where  γ = ∑ i = 1 r   m i α i ∨ (P^(1))^(gamma)=prod_(i=1)^(r)((P^(1))^(m_(i))//S_(m_(i)))quad" where "gamma=sum_(i=1)^(r)m_(i)alpha_(i)^(vv)\left(\mathbb{P}^{1}\right)^{\gamma}=\prod_{i=1}^{r}\left(\left(\mathbb{P}^{1}\right)^{m_{i}} / \mathbb{S}_{m_{i}}\right) \quad \text { where } \gamma=\sum_{i=1}^{r} m_{i} \alpha_{i}^{\vee}(P1)γ=∏i=1r((P1)mi/Smi) where γ=∑i=1rmiαi∨
Here ( P 1 ) γ P 1 γ (P^(1))^(gamma)\left(\mathbb{P}^{1}\right)^{\gamma}(P1)γ records the place where the degree of the genuine map drops in which degree components (without ordering). By adding extra P 1 P 1 P^(1)\mathbb{P}^{1}P1 components and (compatible) maps to
B B B\mathscr{B}B to P 1 P 1 P^(1)\mathbb{P}^{1}P1 in ( f : P 1 B ) Q G ( β γ ) f : P 1 → B ∈ Q ∘ G ( β − γ ) (f:P^(1)rarrB)inQ^(@)_(G)(beta-gamma)\left(f: \mathbb{P}^{1} \rightarrow \mathscr{B}\right) \in \stackrel{\circ}{Q}_{G}(\beta-\gamma)(f:P1→B)∈Q∘G(β−γ) at the places (and total degrees) recorded by ( P 1 ) γ P 1 γ (P^(1))^(gamma)\left(\mathbb{P}^{1}\right)^{\gamma}(P1)γ (for each 0 γ β 0 ≤ γ ≤ β 0 <= gamma <= beta0 \leq \gamma \leq \beta0≤γ≤β ), we obtain a map of topological spaces
π : E B 2 , β b b Q G ( β ) Ï€ : E B 2 , β b b → Q G ( β ) pi:E_(B_(2,beta)^(b))^(b)rarrQ_(G)(beta)\pi: \mathscr{E}_{\mathscr{B}_{2, \beta}^{b}}^{\mathrm{b}} \rightarrow \mathcal{Q}_{G}(\beta)Ï€:EB2,βbb→QG(β)
that is an identity on Q G ( β ) Q ∘ G ( β ) Q^(@)_(G)(beta)\stackrel{\circ}{Q}_{G}(\beta)Q∘G(β). Givental's main lemma asserts that this is a birational morphism of integral algebraic varieties.
Proposition 7.2 ( [ 50 , $ 5.2 ] ) [ 50 , $ 5.2 ] ) [50,$5.2])[50, \$ 5.2])[50,$5.2]). For each β X β ∈ X ∨ beta inX^(vv)\beta \in \mathbb{X}^{\vee}β∈X∨, we have
Q G ( β ) Q G ( e , t β ) Q G ( β ) ≅ Q G e , t β Q_(G)(beta)~=Q_(G)(e,t_(beta))\mathcal{Q}_{G}(\beta) \cong \mathcal{Q}_{G}\left(e, t_{\beta}\right)QG(β)≅QG(e,tβ)
as schemes. In addition, π Ï€ pi\piÏ€ restricts to a birational morphism
π β , w , v : E B β ( w , v ) Q G ( w , v t β ) , w , v W Ï€ β , w , v : E B β ( w , v ) → Q G w , v t β , w , v ∈ W pi_(beta,w,v):EB_(beta)(w,v)rarrQ_(G)(w,vt_(beta)),quad w,v in W\pi_{\beta, w, v}: \mathscr{E} \mathscr{B}_{\beta}(w, v) \rightarrow \mathbb{Q}_{G}\left(w, v t_{\beta}\right), \quad w, v \in Wπβ,w,v:EBβ(w,v)→QG(w,vtβ),w,v∈W
In particular, we have G B β ( w , v ) G B β ( w , v ) ≠ ∅ GB_(beta)(w,v)!=O/\mathscr{G} \mathscr{B}_{\beta}(w, v) \neq \emptysetGBβ(w,v)≠∅ if and only if w 2 v t β w ≤ ∞ 2 v t β w <= (oo)/(2)vt_(beta)w \leq \frac{\infty}{2} v t_{\beta}w≤∞2vtβ, and its dimension is given by the distance between w w www and v t β v t β vt_(beta)v t_{\beta}vtβ with respect to 2 ≤ ∞ 2 <= (oo)/(2)\leq \frac{\infty}{2}≤∞2.
In other words, the Richardson varieties of Q G rat Q G rat  Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat  are precisely the spaces of quasimaps, possibly with additional conditions imposed by the space of stable maps. According to Buch-Chaput-Mihalcea-Perrin [13], the variety E B β ( w , v ) E B β ( w , v ) EB_(beta)(w,v)\mathscr{E} \mathscr{B}_{\beta}(w, v)EBβ(w,v) is irreducible and has rational singularities if it is nonempty. Hence, we find that Q G ( w , v t β ) Q G w , v t β Q_(G)(w,vt_(beta))\mathcal{Q}_{G}\left(w, v t_{\beta}\right)QG(w,vtβ) is irreducible in general. Proposition 7.2 and properties of the maps π β , w , v Ï€ β , w , v pi_(beta,w,v)\pi_{\beta, w, v}πβ,w,v are used in our proof of Theorem 7.1.
Proposition 7.2 implies that Q G ( w , v t β ) Q G w , v t β Q_(G)(w,vt_(beta))\mathcal{Q}_{G}\left(w, v t_{\beta}\right)QG(w,vtβ) is the closure (in Q G ( β ) Q G ( β ) Q_(G)(beta)\mathcal{Q}_{G}(\beta)QG(β) ) of the space of maps from P 1 P 1 P^(1)\mathbb{P}^{1}P1 to B B B\mathscr{B}B such that 0 , P 1 0 , ∞ ∈ P 1 0,oo inP^(1)0, \infty \in \mathbb{P}^{1}0,∞∈P1 land in B ( w ) B ( w ) B(w)\mathscr{B}(w)B(w) and B p ( v ) B ∘ p ( v ) B^(@p)(v)\mathscr{B}^{\circ p}(v)B∘p(v), respectively. By examining the natural map E B β ( w , v ) B β ( w , v ) E B β ( w , v ) → B β ( w , v ) EB_(beta)(w,v)rarrB_(beta)(w,v)\mathscr{E} \mathscr{B}_{\beta}(w, v) \rightarrow \mathscr{B}_{\beta}(w, v)EBβ(w,v)→Bβ(w,v) (obtained by forgetting the map to P 1 P 1 P^(1)\mathbb{P}^{1}P1 ), we obtain:
Corollary 7.3. For all w , v W w , v ∈ W w,v in Ww, v \in Ww,v∈W and 0 β X 0 ≠ β ∈ X ∨ 0!=beta inX^(vv)0 \neq \beta \in \mathbb{X}^{\vee}0≠β∈X∨, we have
dim B β ( w , v ) = dim E B β ( w , v ) 1 if E B β ( w , v ) dim ⁡ B β ( w , v ) = dim ⁡ E B β ( w , v ) − 1  if  E B β ( w , v ) ≠ ∅ dim B_(beta)(w,v)=dim EB_(beta)(w,v)-1quad" if "EB_(beta)(w,v)!=O/\operatorname{dim} \mathscr{B}_{\beta}(w, v)=\operatorname{dim} \mathscr{E} \mathscr{B}_{\beta}(w, v)-1 \quad \text { if } \mathscr{E} \mathscr{B}_{\beta}(w, v) \neq \emptysetdim⁡Bβ(w,v)=dim⁡EBβ(w,v)−1 if EBβ(w,v)≠∅
and B β ( w , v ) B β ( w , v ) ≠ ∅ B_(beta)(w,v)!=O/\mathscr{B}_{\beta}(w, v) \neq \emptysetBβ(w,v)≠∅ if and only if E B β ( w , v ) E B β ( w , v ) ≠ ∅ EB_(beta)(w,v)!=O/\mathcal{E} \mathscr{B}_{\beta}(w, v) \neq \emptysetEBβ(w,v)≠∅. Moreover, we have
B β ( w , v ) and dim B β ( w , v ) = 0 B β ( w , v ) ≠ ∅  and  dim ⁡ B β ( w , v ) = 0 B_(beta)(w,v)!=O/quad" and "quad dim B_(beta)(w,v)=0\mathscr{B}_{\beta}(w, v) \neq \emptyset \quad \text { and } \quad \operatorname{dim} \mathscr{B}_{\beta}(w, v)=0Bβ(w,v)≠∅ and dim⁡Bβ(w,v)=0
if and only if w 2 v t β w ≤ ∞ 2 v t β w <= (oo)/(2)vt_(beta)w \leq \frac{\infty}{2} v t_{\beta}w≤∞2vtβ are adjacent with respect to 2 ≤ ∞ 2 <= (oo)/(2)\leq \frac{\infty}{2}≤∞2. In such a case, B β ( w , v ) B β ( w , v ) B_(beta)(w,v)\mathscr{B}_{\beta}(w, v)Bβ(w,v) is a point.
Thanks to the dimension axiom in quantum correlators [54, (2.5)], Corollary 7.3 describes which (primary) two-point cohomological Gromov-Witten invariant of B B B\mathscr{B}B with respect to the Schubert bases is nonzero (we can also tell its exact value). By the divisor axiom [54, $2.2.4] and the classical Chevalley formula [16], we find the Chevalley formula in quantum cohomology of B B B\mathscr{B}B from this [29]. This clarifies the role of Q G ( w , v t β ) Q G w , v t β Q_(G)(w,vt_(beta))\mathcal{Q}_{G}\left(w, v t_{\beta}\right)QG(w,vtβ) in the study of quantum cohomology of B B B\mathscr{B}B from our perspective.
Theorem 7.4 ([47]). Let β X β ∈ X ∨ beta inX^(vv)\beta \in \mathbb{X}^{\vee}β∈X∨ and w , v W w , v ∈ W w,v in Ww, v \in Ww,v∈W. The variety Q G ( w , v t β ) Q G w , v t β Q_(G)(w,vt_(beta))Q_{G}\left(w, v t_{\beta}\right)QG(w,vtβ) has rational singularities.
Theorem 7.4 is proved by Braverman-Finkelberg [ 9 , 10 ] [ 9 , 10 ] [9,10][9,10][9,10] for the case w = e , v = w 0 w = e , v = w 0 w=e,v=w_(0)w=e, v=w_{0}w=e,v=w0 by an analysis of Zastava spaces, which does not extend to general w , v w , v w,vw, vw,v. Theorem 7.4 is the most subtle technical point in [47] and its induction steps become possible by Theorem 7.1.

8. K K KKK-THEORETIC PETERSON ISOMORPHISM

We follow the setting of the previous section with k = C k = C k=C\mathbb{k}=\mathbb{C}k=C. We understand that the K K KKK-groups appearing here contain a suitable class of line bundles supported on subvarieties equipped with some group actions, and its scalar is extended from Z Z Z\mathbb{Z}Z to C C C\mathbb{C}C. Let Gr G := G ( ( z ) ) / G [ [ z ] ] Gr G := G ( ( z ) ) / G [ [ z ] ] Gr_(G):=G((z))//G[[z]]\operatorname{Gr}_{G}:=G((z)) / G \llbracket z \rrbracketGrG:=G((z))/G[[z]] be the affine Grassmannian of G G GGG. The set of I I I\mathbf{I}I-orbits in G r G G r G Gr_(G)\mathrm{Gr}_{G}GrG is in bijection with X X ∨ X^(vv)\mathbb{X}^{\vee}X∨, while the set of G [ [ z ] ] G [ [ z ] ] G[[z]]G \llbracket z \rrbracketG[[z]]-orbits of G r G G r G Gr_(G)\mathrm{Gr}_{G}GrG is in bijection with X < X X < ∨ ⊂ X ∨ X_( < )^(vv)subX^(vv)\mathbb{X}_{<}^{\vee} \subset \mathbb{X}^{\vee}X<∨⊂X∨ formed by the set of antidominant coroots. For β X β ∈ X ∨ beta inX^(vv)\beta \in \mathbb{X}^{\vee}β∈X∨, we set Gr G ( β ) Gr G Gr G ⁡ ( β ) ⊂ Gr G Gr_(G)(beta)subGr_(G)\operatorname{Gr}_{G}(\beta) \subset \operatorname{Gr}_{G}GrG⁡(β)⊂GrG as the corresponding I-orbit and set Gr G ( β ) := ¯ G r G ( β ) Gr G Gr G ⁡ ( β ) := ∘ ¯ G r G ( β ) ⊂ Gr G Gr_(G)(beta):= bar(@)_(Gr_(G))(beta)subGr_(G)\operatorname{Gr}_{G}(\beta):=\bar{\circ}_{\mathrm{Gr}_{G}}(\beta) \subset \operatorname{Gr}_{G}GrG⁡(β):=∘¯GrG(β)⊂GrG. We normalize so that Gr G ( β ) Gr G ⁡ ( β ) Gr_(G)(beta)\operatorname{Gr}_{G}(\beta)GrG⁡(β) is G G GGG-stable when β X < β ∈ X < ∨ beta inX_( < )^(vv)\beta \in \mathbb{X}_{<}^{\vee}β∈X<∨, and we have dim Gr G ( β ) = 2 | β | dim ⁡ Gr G ⁡ ( β ) = − 2 | β | dim Gr_(G)(beta)=-2|beta|\operatorname{dim} \operatorname{Gr}_{G}(\beta)=-2|\beta|dim⁡GrG⁡(β)=−2|β| in such a case, where | β | := i = 1 r β ( ϖ i ) | β | := ∑ i = 1 r   β Ï– i |beta|:=sum_(i=1)^(r)beta(Ï–_(i))|\beta|:=\sum_{i=1}^{r} \beta\left(\varpi_{i}\right)|β|:=∑i=1rβ(Ï–i).
We define
K T ( Gr G ) := β X K T ( Gr G ( β ) ) and K G ( Gr G ) := β X K G ( Gr G ( β ) ) K T Gr G := ⋃ β ∈ X ∨   K T Gr G ⁡ ( β )  and  K G Gr G := ⋃ β ∈ X ∨   K G Gr G ⁡ ( β ) K_(T)(Gr_(G)):=uuu_(beta inX^(vv))K_(T)(Gr_(G)(beta))quad" and "quadK_(G)(Gr_(G)):=uuu_(beta inXvv)K_(G)(Gr_(G)(beta))K_{T}\left(\operatorname{Gr}_{G}\right):=\bigcup_{\beta \in \mathbb{X}^{\vee}} K_{T}\left(\operatorname{Gr}_{G}(\beta)\right) \quad \text { and } \quad K_{G}\left(\operatorname{Gr}_{G}\right):=\bigcup_{\beta \in \mathbb{X} \vee} K_{G}\left(\operatorname{Gr}_{G}(\beta)\right)KT(GrG):=⋃β∈X∨KT(GrG⁡(β)) and KG(GrG):=⋃β∈X∨KG(GrG⁡(β))
These spaces are equipped with the convolution product, defined by the diagram
G r G × G r G p G ( ( z ) ) × Gr G q G ( ( z ) ) × × I G r G mult G r G G r G × G r G ← p G ( ( z ) ) × Gr G → q G ( ( z ) ) × × I G r G →  mult  G r G Gr_(G)xxGr_(G)larr^(p)G((z))xxGr_(G)rarr"q"G((z))xxxx_(I)Gr_(G)rarr"" mult ""Gr_(G)\mathrm{Gr}_{G} \times \mathrm{Gr}_{G} \stackrel{p}{\leftarrow} G((z)) \times \operatorname{Gr}_{G} \xrightarrow{q} G((z)) \times \times_{\mathbf{I}} \mathrm{Gr}_{G} \xrightarrow{\text { mult }} \mathrm{Gr}_{G}GrG×GrG←pG((z))×GrG→qG((z))××IGrG→ mult GrG
as follows: For all cycles a , b K T ( G r G ) K I ( G r G ) a , b ∈ K T G r G ≅ K I G r G a,b inK_(T)(Gr_(G))~=K_(I)(Gr_(G))a, b \in K_{T}\left(\mathrm{Gr}_{G}\right) \cong K_{\mathrm{I}}\left(\mathrm{Gr}_{G}\right)a,b∈KT(GrG)≅KI(GrG), we find a left I I I\mathbf{I}I-equivariant class ( a , b ) ( a , b ) (a,b)(a, b)(a,b) on G ( ( z ) ) × I G r G G ( ( z ) ) × I G r G G((z))xx_(I)Gr_(G)G((z)) \times_{\mathrm{I}} \mathrm{Gr}_{G}G((z))×IGrG such that
p ( a b ) = q ( a , b ) p ∗ ( a ⊠ b ) = q ∗ ( a , b ) p^(**)(a⊠b)=q^(**)(a,b)p^{*}(a \boxtimes b)=q^{*}(a, b)p∗(a⊠b)=q∗(a,b)
and set
a b := i 0 ( 1 ) i [ R i mult ( a , b ) ] K I ( Gr G ) a ⊙ ′ b := ∑ i ≥ 0   ( − 1 ) i R i mult ∗ ⁡ ( a , b ) ∈ K I Gr G ao.^(')b:=sum_(i >= 0)(-1)^(i)[R^(i)mult_(**)(a,b)]inK_(I)(Gr_(G))a \odot^{\prime} b:=\sum_{i \geq 0}(-1)^{i}\left[\mathbb{R}^{i} \operatorname{mult}_{*}(a, b)\right] \in K_{\mathbf{I}}\left(\operatorname{Gr}_{G}\right)a⊙′b:=∑i≥0(−1)i[Rimult∗⁡(a,b)]∈KI(GrG)
This yields an associative product structure on K T ( G r G ) K T G r G K_(T)(Gr_(G))K_{T}\left(\mathrm{Gr}_{G}\right)KT(GrG) that contains a zero divisor. If we restrict ourselves to K G ( Gr G ) K G Gr G K_(G)(Gr_(G))K_{G}\left(\operatorname{Gr}_{G}\right)KG(GrG), then the algebra structure given by ⊙ ′ o.^(')\odot^{\prime}⊙′ becomes commutative and integrally closed. Using an isomorphism K T ( p t ) K G ( p t ) K G ( G r G ) K T ( G r G ) K T ( p t ) ⊗ K G ( p t ) K G G r G ≅ K T G r G K_(T)(pt)ox_(K_(G)(pt))K_(G)(Gr_(G))~=K_(T)(Gr_(G))K_{T}(\mathrm{pt}) \otimes_{K_{G}(\mathrm{pt})} K_{G}\left(\mathrm{Gr}_{G}\right) \cong K_{T}\left(\mathrm{Gr}_{G}\right)KT(pt)⊗KG(pt)KG(GrG)≅KT(GrG) of K T ( p t ) K T ( p t ) K_(T)(pt)K_{T}(\mathrm{pt})KT(pt)-modules, we find a multiplication ⊙ o.\odot⊙ of K T ( G r G ) K T G r G K_(T)(Gr_(G))K_{T}\left(\mathrm{Gr}_{G}\right)KT(GrG) that extends ⊙ ′ o.^(')\odot^{\prime}⊙′ on K G ( G r G ) K G G r G K_(G)(Gr_(G))K_{G}\left(\mathrm{Gr}_{G}\right)KG(GrG) as a K T ( p t ) K T ( p t ) K_(T)(pt)K_{T}(\mathrm{pt})KT(pt)-algebra. This product ⊙ o.\odot⊙ coincides with a K K KKK-theoretic analogue of the Pontrjagin product (by the calculations in [ 47 , $ 2.2 ] [ 47 , $ 2.2 ] [47,$2.2][47, \$ 2.2][47,$2.2] ). In addition, we have
[ O Gr G ( β + γ ) ] = [ O Gr G ( β ) ] [ O Gr G ( γ ) ] for β , γ X < O Gr G ⁡ ( β + γ ) = O Gr G ⁡ ( β ) ⊙ O Gr G ⁡ ( γ )  for  β , γ ∈ X < ∨ [O_(Gr_(G)(beta+gamma))]=[O_(Gr_(G)(beta))]o.[O_(Gr_(G)(gamma))]quad" for "quad beta,gamma inX_( < )^(vv)\left[\mathcal{O}_{\operatorname{Gr}_{G}(\beta+\gamma)}\right]=\left[\mathcal{O}_{\operatorname{Gr}_{G}(\beta)}\right] \odot\left[\mathcal{O}_{\operatorname{Gr}_{G}(\gamma)}\right] \quad \text { for } \quad \beta, \gamma \in \mathbb{X}_{<}^{\vee}[OGrG⁡(β+γ)]=[OGrG⁡(β)]⊙[OGrG⁡(γ)] for β,γ∈X<∨
This yields a multiplicative system in K T ( G r G ) K T G r G K_(T)(Gr_(G))K_{T}\left(\mathrm{Gr}_{G}\right)KT(GrG), whose localization is denoted by K T ( Gr G ) l o c K T Gr G l o c K_(T)(Gr_(G))_(loc)K_{T}\left(\operatorname{Gr}_{G}\right)_{\mathrm{loc}}KT(GrG)loc.
The (localized) small T T TTT-equivariant quantum K K KKK-group of B B B\mathscr{B}B is defined as a vector space
q K T ( B ) l o c := K T ( B ) C X ( K T ( B ) C C H 2 ( B , Z ) ) q K T ( B ) l o c := K T ( B ) ⊗ C X ∨ ≡ K T ( B ) ⊗ C C H 2 ( B , Z ) qK_(T)(B)_(loc):=K_(T)(B)oxCX^(vv)quad(-=K_(T)(B)oxCCH_(2)(B,Z))q K_{T}(\mathscr{B})_{\mathrm{loc}}:=K_{T}(\mathscr{B}) \otimes \mathbb{C} \mathbb{X}^{\vee} \quad\left(\equiv K_{T}(\mathscr{B}) \otimes \mathbb{C} \mathbb{C} H_{2}(\mathscr{B}, \mathbb{Z})\right)qKT(B)loc:=KT(B)⊗CX∨(≡KT(B)⊗CCH2(B,Z))
We denote the variable corresponding to β X β ∈ X ∨ beta inX^(vv)\beta \in \mathbb{X}^{\vee}β∈X∨ as Q β Q β Q^(beta)Q^{\beta}Qβ. The quantum K K KKK-theoretic product ⋆ ***\star⋆ is a binary operation on q K T ( B ) l o c q K T ( B ) l o c qK_(T)(B)_(loc)q K_{T}(\mathscr{B})_{\mathrm{loc}}qKT(B)loc, defined by Givental [33] and Lee [61], whose value (a priori) belongs to a completion of q K T ( B ) loc q K T ( B ) loc  qK_(T)(B)_("loc ")q K_{T}(\mathscr{B})_{\text {loc }}qKT(B)loc . It is one of the consequence of our analysis that ⋆ ***\star⋆ preserves q K T ( B ) loc q K T ( B ) loc  qK_(T)(B)_("loc ")q K_{T}(\mathscr{B})_{\text {loc }}qKT(B)loc . This is usually referred to as the finiteness of the quantum K K KKK-theoretic product (for B B B\mathscr{B}B ) in the literature [1,13], and is one of the most fundamental questions in the study of q K T ( B ) q K T ( B ) qK_(T)(B)q K_{T}(\mathscr{B})qKT(B). Lam-Li-Mihalcea-Shimozono [58] conjectured that:
Theorem 8.1 ([47]). We have an isomorphism of commutative algebras
K T ( G r G ) l o c q K T ( B ) l o c K T G r G l o c ⇆ ≅ q K T ( B ) l o c K_(T)(Gr_(G))_(loc)⇆^(~=)qK_(T)(B)_(loc)K_{T}\left(\mathrm{Gr}_{G}\right)_{\mathrm{loc}} \stackrel{\cong}{\leftrightarrows} q K_{T}(\mathscr{B})_{\mathrm{loc}}KT(GrG)loc⇆≅qKT(B)loc
such that
[ O Gr G ( w β ) ] [ O Gr G ( γ ) ] 1 [ O B ( w ) ] Q β γ O Gr G ⁡ ( w β ) ⊙ O Gr G ⁡ ( γ ) − 1 ↦ O B ( w ) Q β − γ [O_(Gr_(G)(w beta))]o.[O_(Gr_(G)(gamma))]^(-1)|->[O_(B(w))]Q^(beta-gamma)\left[\mathcal{O}_{\operatorname{Gr}_{G}(w \beta)}\right] \odot\left[\mathcal{O}_{\operatorname{Gr}_{G}(\gamma)}\right]^{-1} \mapsto\left[\mathcal{O}_{\mathcal{B}(w)}\right] Q^{\beta-\gamma}[OGrG⁡(wβ)]⊙[OGrG⁡(γ)]−1↦[OB(w)]Qβ−γ
for β , γ X < β , γ ∈ X < ∨ beta,gamma inX_( < )^(vv)\beta, \gamma \in \mathbb{X}_{<}^{\vee}β,γ∈X<∨ such that β ( ϖ i ) < 0 β Ï– i < 0 beta(Ï–_(i)) < 0\beta\left(\varpi_{i}\right)<0β(Ï–i)<0 for every 1 i r 1 ≤ i ≤ r 1 <= i <= r1 \leq i \leq r1≤i≤r.
Note that a presentation of the ring q K T ( B ) q K T ( B ) qK_(T)(B)q K_{T}(\mathscr{B})qKT(B) for G = SL ( n ) G = SL ⁡ ( n ) G=SL(n)G=\operatorname{SL}(n)G=SL⁡(n) can be read-off from Givental-Lee [34], and a presentation of the ring K T ( G r G ) K T G r G K_(T)(Gr_(G))K_{T}\left(\mathrm{Gr}_{G}\right)KT(GrG) is obtained in BezrukavnikovFinkelberg-Mirković [6]. However, these are not enough to yield Theorem 8.1 (for G = S L ( n ) ) G = S L ( n ) ) G=SL(n))G=\mathrm{SL}(n))G=SL(n)) as the correspondence between Schubert bases is unclear.
We have an action of the nilpotent version H nil H nil  H^("nil ")\mathscr{H}^{\text {nil }}Hnil  of the double affine Hecke algebra (associated to G ) G ) G)G)G) on K T ( G r G ) K T G r G K_(T)(Gr_(G))K_{T}\left(\mathrm{Gr}_{G}\right)KT(GrG), coming from Kostant-Kumar [55]. In [47], we defined the T T TTT-equivariant K K KKK-group K T ( Q G rat ) K T Q G rat  K_(T)(Q_(G)^("rat "))K_{T}\left(\mathbf{Q}_{G}^{\text {rat }}\right)KT(QGrat ) of Q G rat Q G rat  Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat  based on the construction of the ( T × G m rot ) T × G m rot  (T xxG_(m)^("rot "))\left(T \times \mathbb{G}_{m}^{\text {rot }}\right)(T×Gmrot ) equivariant K K KKK-group of Q G rat Q G rat  Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat  in [52]. The I I I\mathbf{I}I-action on Q G rat Q G rat  Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat  induces a H H nil H H nil  HH^("nil ")\mathscr{H H}^{\text {nil }}HHnil -action on K T ( Q G rat ) K T Q G rat  K_(T)(Q_(G)^("rat "))K_{T}\left(\mathbf{Q}_{G}^{\text {rat }}\right)KT(QGrat ).
The object K T ( Q G rat ) K T Q G rat  K_(T)(Q_(G)^("rat "))K_{T}\left(\mathbf{Q}_{G}^{\text {rat }}\right)KT(QGrat ) needs a completion in order to admit an action of the line bundle twists by O Q G rat ( λ ) ( λ X ) O Q G rat  ( λ ) ( λ ∈ X ) O_(Q_(G)^("rat "))(lambda)(lambda inX)\mathcal{O}_{\mathbf{Q}_{G}^{\text {rat }}}(\lambda)(\lambda \in \mathbb{X})OQGrat (λ)(λ∈X). It reflects the fact that the right-hand side of Theorem 5.3 (i.e., a global Weyl module) is infinite-dimensional in general, and hence the effect of O Q G rat ( ϖ i ) ( 1 i r ) ⊗ O Q G rat  Ï– i ( 1 ≤ i ≤ r ) oxO_(Q_(G)^("rat "))(Ï–_(i))(1 <= i <= r)\otimes \mathcal{O}_{\mathbf{Q}_{G}^{\text {rat }}}\left(\varpi_{i}\right)(1 \leq i \leq r)⊗OQGrat (Ï–i)(1≤i≤r) requires infinitely many terms to describe.
Our main idea in the proof of Theorem 8.1 is to put Q G rat Q G rat  Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat  into the picture:
Theorem 8.2 ([47, THEOREM c]). We have a commutative diagram
that respects the Schubert bases in each object. In addition, the map Ψ Î¨ Psi\PsiΨ is an embedding of representations of H H nil H H nil  HH^("nil ")\mathscr{H} \mathscr{H}^{\text {nil }}HHnil , and the map Ψ Î¨ Psi\PsiΨ intertwines the tensor product with O Q G rat ( ϖ i ) O Q G rat  − Ï– i O_(Q_(G)^("rat "))(-Ï–_(i))\mathcal{O}_{\mathbf{Q}_{G}^{\text {rat }}}\left(-\varpi_{i}\right)OQGrat (−ϖi) in K T ( Q G rat ) K T Q G rat  K_(T)(Q_(G)^("rat "))K_{T}\left(\mathbf{Q}_{G}^{\text {rat }}\right)KT(QGrat ) and the quantum product of O B ( ϖ i ) O B − Ï– i O_(B)(-Ï–_(i))\mathcal{O}_{\mathcal{B}}\left(-\varpi_{i}\right)OB(−ϖi) on q K T ( B ) l o c q K T ( B ) l o c qK_(T)(B)_(loc)q K_{T}(\mathscr{B})_{\mathrm{loc}}qKT(B)loc for each 1 i r 1 ≤ i ≤ r 1 <= i <= r1 \leq i \leq r1≤i≤r.
The completion of K T ( Q G rat ) K T Q G rat  K_(T)(Q_(G)^("rat "))K_{T}\left(\mathbf{Q}_{G}^{\text {rat }}\right)KT(QGrat ) is compatible with the standard completion of q K T ( B ) q K T ( B ) qK_(T)(B)q K_{T}(\mathscr{B})qKT(B) via the map Ψ Î¨ Psi\PsiΨ. Theorem 8.2 implies that the inverse of the operation O B ( ϖ i ) ⋆ O B − Ï– i ***O_(B)(-Ï–_(i))\star \mathcal{O}_{\mathcal{B}}\left(-\varpi_{i}\right)⋆OB(−ϖi) makes sense only after the completion of q K T ( B ) l o c q K T ( B ) l o c qK_(T)(B)_(loc)q K_{T}(\mathscr{B})_{\mathrm{loc}}qKT(B)loc.
Since the quantum K K KKK-theoretic correlators (see [ 33 , 61 ] [ 33 , 61 ] [33,61][33,61][33,61] ) satisfy neither the dimension axiom nor divisor axiom as in the theory of quantum cohomology, the proof of Theorem 8.2
must be necessarily different from Corollary 7.3. Our construction of the map Ψ Î¨ Psi\PsiΨ is based on the following two observations:
  • an interpretation of the ( G m G m G_(m)\mathbb{G}_{m}Gm-equivariant) quantum K K KKK-theoretic correlator
(8.1) χ ( Q ( w , w 0 t β ) , O Q ( w , w 0 t β ) ( λ ) ) = χ ( B B β ( w , w 0 ) , π β , w , w 0 O Q ( w , w 0 t β ) ( λ ) ) (8.1) χ Q w , w 0 t β , O Q w , w 0 t β ( λ ) = χ B B β w , w 0 , Ï€ β , w , w 0 ∗ O Q w , w 0 t β ( λ ) {:(8.1)chi(Q(w,w_(0)t_(beta)),O_(Q(w,w_(0)t_(beta)))(lambda))=chi(BB_(beta)(w,w_(0)),pi_(beta,w,w_(0))^(**)O_(Q(w,w_(0)t_(beta)))(lambda)):}\begin{equation*} \chi\left(\mathcal{Q}\left(w, w_{0} t_{\beta}\right), \mathcal{O}_{\mathscr{Q}\left(w, w_{0} t_{\beta}\right)}(\lambda)\right)=\chi\left(\mathscr{B} \mathscr{B}_{\beta}\left(w, w_{0}\right), \pi_{\beta, w, w_{0}}^{*} \mathcal{O}_{\mathbb{Q}\left(w, w_{0} t_{\beta}\right)}(\lambda)\right) \tag{8.1} \end{equation*}(8.1)χ(Q(w,w0tβ),OQ(w,w0tβ)(λ))=χ(BBβ(w,w0),πβ,w,w0∗OQ(w,w0tβ)(λ))
valued in C [ T ] [ q ± 1 ] = C [ T × G m ] C [ T ] q ± 1 = C T × G m C[T][q^(+-1)]=C[T xxG_(m)]\mathbb{C}[T]\left[q^{ \pm 1}\right]=\mathbb{C}\left[T \times \mathbb{G}_{m}\right]C[T][q±1]=C[T×Gm], for each w W , β X , λ X + w ∈ W , β ∈ X ∨ , λ ∈ X + w in W,beta inX^(vv),lambda inX_(+)w \in W, \beta \in \mathbb{X}^{\vee}, \lambda \in \mathbb{X}_{+}w∈W,β∈X∨,λ∈X+;
  • an interpretation of its asymptotic behavior
(8.2) lim β χ ( Q ( w , w 0 t β ) , O Q ( w , w 0 t β ) ( λ ) ) = χ ( Q G ( w ) , O Q G ( w ) ( λ ) ) C ( ( q 1 ) ) [ T ] (8.2) lim β → ∞   χ Q w , w 0 t β , O Q w , w 0 t β ( λ ) = χ Q G ( w ) , O Q G ( w ) ( λ ) ∈ C q − 1 [ T ] {:(8.2)lim_(beta rarr oo)chi(Q(w,w_(0)t_(beta)),O_(Q(w,w_(0)t_(beta)))(lambda))=chi(Q_(G)(w),O_(Q_(G)(w))(lambda))inC((q^(-1)))[T]:}\begin{equation*} \lim _{\beta \rightarrow \infty} \chi\left(\mathcal{Q}\left(w, w_{0} t_{\beta}\right), \mathcal{O}_{\mathbb{Q}\left(w, w_{0} t_{\beta}\right)}(\lambda)\right)=\chi\left(\mathbf{Q}_{G}(w), \mathcal{O}_{\mathbf{Q}_{G}(w)}(\lambda)\right) \in \mathbb{C}\left(\left(q^{-1}\right)\right)[T] \tag{8.2} \end{equation*}(8.2)limβ→∞χ(Q(w,w0tβ),OQ(w,w0tβ)(λ))=χ(QG(w),OQG(w)(λ))∈C((q−1))[T]
for each w W , λ X + w ∈ W , λ ∈ X + w in W,lambda inX_(+)w \in W, \lambda \in \mathbb{X}_{+}w∈W,λ∈X+as an element of K T ( Q G rat ) K T Q G rat  K_(T)(Q_(G)^("rat "))K_{T}\left(\mathbf{Q}_{G}^{\text {rat }}\right)KT(QGrat ).
Here we can further interpret χ ( E B β ( w , w 0 ) , π β , w , w 0 O Q ( w , w 0 t β ) ( λ ) ) χ E B β w , w 0 , Ï€ β , w , w 0 ∗ O Q w , w 0 t β ( λ ) chi(E_(B_(beta))(w,w_(0)),pi_(beta,w,w_(0))^(**)O_(Q(w,w_(0)t_(beta)))(lambda))\chi\left(\mathscr{E}_{\mathscr{B}_{\beta}}\left(w, w_{0}\right), \pi_{\beta, w, w_{0}}^{*} \mathcal{O}_{\mathscr{Q}\left(w, w_{0} t_{\beta}\right)}(\lambda)\right)χ(EBβ(w,w0),πβ,w,w0∗OQ(w,w0tβ)(λ)) using the shift operators of line bundles in quantum K K KKK-theory [35, PROPOSITION 2.13], and hence we obtain an (abstract) presentation of q K T ( B ) q K T ( B ) qK_(T)(B)q K_{T}(\mathscr{B})qKT(B) from (8.1) by the reconstruction theorem [35, PROPOSITION 2.12]. The identity (8.1) is a consequence of Theorem 7.4, and (8.2) is a consequence of compatible Frobenius splitting properties of Q G ( w , v ) s Q G ( w , v ) s Q_(G)(w,v)s\mathcal{Q}_{G}(w, v) \mathrm{s}QG(w,v)s and Q G rat Q G rat  Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat  in char k > 2 k > 2 k > 2\mathbb{k}>2k>2 (see the explanation about the proof of Theorem 7.1).
There is a noncommutative version of Theorem 8.2, meaning that we include G m rot G m rot  G_(m)^("rot ")\mathbb{G}_{m}^{\text {rot }}Gmrot  (the variable " q q qqq " above) in each item [49].

9. FUNCTORIALITY OF QUANTUM K K KKK-GROUPS

We continue to work in the setting as in the previous section. In [50], we have presented analogues of Theorems 5.2, 5.3, and 7.1 for partial flag manifolds of G G GGG. Let us find a standard parabolic subgroup B P G B ⊂ P ⊂ G B sub P sub GB \subset P \subset GB⊂P⊂G and consider B P := G / P B P := G / P B_(P):=G//P\mathscr{B}_{P}:=G / PBP:=G/P. Our parabolic version of the semi-infinite flag manifold Q G , P rat Q G , P rat  Q_(G,P)^("rat ")\mathbf{Q}_{G, P}^{\text {rat }}QG,Prat  has its set of k k k\mathbb{k}k-valued points G ( ( z ) ) / ( T [ P , P ] ( ( z ) ) ) G ( ( z ) ) / ( T ⋅ [ P , P ] ( ( z ) ) ) G((z))//(T*[P,P]((z)))G((z)) /(T \cdot[P, P]((z)))G((z))/(T⋅[P,P]((z))). The fiber of the natural map
π P : Q G rat Q G , P rat Ï€ P : Q G rat  → Q G , P rat  pi_(P):Q_(G)^("rat ")rarrQ_(G,P)^("rat ")\pi_{P}: \mathbf{Q}_{G}^{\text {rat }} \rightarrow \mathbf{Q}_{G, P}^{\text {rat }}Ï€P:QGrat →QG,Prat 
is isomorphic to the semi-infinite flag manifold of [ L , L ] [ L , L ] [L,L][L, L][L,L], where L P L ⊂ P L sub PL \subset PL⊂P is the maximal semisimple subgroup of P P PPP that contains T T TTT (the standard Levi subgroup). We also have the higher cohomology vanishing of equivariant line bundles on Q G , P rat Q G , P rat  Q_(G,P)^("rat ")\mathbf{Q}_{G, P}^{\text {rat }}QG,Prat  (or rather π P ( Q G ) ) Ï€ P Q G {:pi_(P)(Q_(G)))\left.\pi_{P}\left(\mathbf{Q}_{G}\right)\right)Ï€P(QG)) as in Theorem 5.3. These are enough to yield a morphism
K T × G m rot ( Q G rat ) K T × G m rot ( Q G , P rat ) K T × G m rot  Q G rat  → K T × G m rot  Q G , P rat  K_(T xxG_(m)^("rot "))(Q_(G)^("rat "))rarrK_(T xxG_(m)^("rot "))(Q_(G,P)^("rat "))K_{T \times \mathbb{G}_{m}^{\text {rot }}}\left(\mathbf{Q}_{G}^{\text {rat }}\right) \rightarrow K_{T \times \mathbb{G}_{m}^{\text {rot }}}\left(\mathbf{Q}_{G, P}^{\text {rat }}\right)KT×Gmrot (QGrat )→KT×Gmrot (QG,Prat )
obtained by the push-forward by π P Ï€ P pi_(P)\pi_{P}Ï€P (up to technical reservations neglected here and below).
By transferring Theorem 7.4 to Richardson varieties of Q G , P rat Q G , P rat  Q_(G,P)^("rat ")\mathbf{Q}_{G, P}^{\text {rat }}QG,Prat , we find a map
Ψ P : q K T ( B P ) l o c K T ( Q G , P r a t ) Ψ P : q K T B P l o c → K T Q G , P r a t Psi_(P):qK_(T)(B_(P))_(loc)rarrK_(T)(Q_(G,P)^(rat))\Psi_{P}: q K_{T}\left(\mathscr{B}_{P}\right)_{\mathrm{loc}} \rightarrow K_{T}\left(\mathbf{Q}_{G, P}^{\mathrm{rat}}\right)ΨP:qKT(BP)loc→KT(QG,Prat)
that intertwines appropriate line bundle twists (and analogous quantum multiplications). This yields a diagram
where we set Q G , P := π P ( Q G ) Q G , P := Ï€ P Q G Q_(G,P):=pi_(P)(Q_(G))\mathbf{Q}_{G, P}:=\pi_{P}\left(\mathbf{Q}_{G}\right)QG,P:=Ï€P(QG).
The resulting map q K T ( B ) q K T ( B P ) q K T ( B ) → q K T B P qK_(T)(B)rarr qK_(T)(B_(P))q K_{T}(\mathscr{B}) \rightarrow q K_{T}\left(\mathscr{B}_{P}\right)qKT(B)→qKT(BP) is, in fact, an algebra map [48], and is easy to describe. Note that we cannot have an analogous map between ordinary K K KKK-groups because of the higher direct images. It turns out this map sends Q α i Q α i ∨ Q^(alpha_(i)^(vv))Q^{\alpha_{i}^{\vee}}Qαi∨ to 1 for a simple coroot α i α i ∨ alpha_(i)^(vv)\alpha_{i}^{\vee}αi∨ belonging to L L LLL, and hence is not compatible with a naive generalization of the corresponding map in the Peterson isomorphism in homology [59].
We also have a restriction map q K T ( B ) q K T ( B L ) q K T ( B ) → q K T B L qK_(T)(B)rarr qK_(T)(B^(L))q K_{T}(\mathscr{B}) \rightarrow q K_{T}\left(\mathcal{B}^{L}\right)qKT(B)→qKT(BL), where B L := L / ( L B ) B L := L / ( L ∩ B ) B^(L):=L//(L nn B)\mathcal{B}^{L}:=L /(L \cap B)BL:=L/(L∩B) is the flag manifold of a standard Levi subgroup. This map extends to algebra maps [45]
K G × G m root ( G r G ) K L × G m roo ( G r L ) K T × G m room ( G r T ) K G × G m root  G r G → K L × G m roo  G r L → K T × G m room  G r T K_(G xxG_(m)^("root "))(Gr_(G))rarrK_(L xxG_(m)^("roo "))(Gr_(L))rarrK_(T xxG_(m)^("room "))(Gr_(T))K_{G \times \mathbb{G}_{m}^{\text {root }}}\left(\mathrm{Gr}_{G}\right) \rightarrow K_{L \times \mathbb{G}_{m}^{\text {roo }}}\left(\mathrm{Gr}_{L}\right) \rightarrow K_{T \times \mathbb{G}_{m}^{\text {room }}}\left(\mathrm{Gr}_{T}\right)KG×Gmroot (GrG)→KL×Gmroo (GrL)→KT×Gmroom (GrT)
anticipated in Finkelberg and Tsymbaliuk [23].

10. SOME PERSPECTIVES

Compared with the theory of flag manifolds, many precise results and constructions for Q G rat Q G rat  Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat  are still missing. The most accessible set of problems might be to spell out analogues of numerous explicit formulas in classical Schubert calculus purely combinatorially by admitting geometric conclusions from [ 3 , 45 , 47 , 48 , 52 ] [ 3 , 45 , 47 , 48 , 52 ] [3,45,47,48,52][3,45,47,48,52][3,45,47,48,52] partly explained in the previous two sections. We close this note by briefly discussing some of other problems.

10.1. Categorifications of the coordinate rings

The homogeneous coordinate rings of Schubert varieties of a usual flag manifold, that are B B BBB-stable quotient rings of (2.1), can be seen as the Grothendieck groups of suitable categories equipped with cluster structures ([60]; see also Section 3.1). Hence, it is natural to expect categorifications of the homogeneous coordinate rings of Q G rat ( w ) Q G rat  ( w ) Q_(G)^("rat ")(w)\mathbf{Q}_{G}^{\text {rat }}(w)QGrat (w) and B C thick B C thick  B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }}BCthick . See also [21] and [43] for related problems and partial answers.

10.2. Peterson isomorphism in quantum cohomology

The Peterson isomorphism in quantum cohomology [59,74] is an analogue of Theorem 8.1 for homology. We may apply Corollary 7.3 to [69] (that is an essential ingredient in [59]) to utilize Q G rat Q G rat  Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat  in its proof (that looks similar to the original strategy in [74]). However, we do not know an analogue of Theorem 8.2 as we lack a proper definition of H ( Q G ) H ∙ Q G H^(∙)(Q_(G))H^{\bullet}\left(\mathbf{Q}_{G}\right)H∙(QG).

10.3. Constructible sheaves on semi-infinite flags

In representation-theoretic analysis on B B B\mathscr{B}B, we sometimes encounter constructible sheaves that are not N N NNN-equivariant. Also, we want some notion of (co)homology of Q G Q G Q_(G)\mathbf{Q}_{G}QG in
Section 10.2. Therefore, it is desirable to understand constructible sheaves on Q G Q G Q_(G)\mathbf{Q}_{G}QG following [7]. The resulting objects should have connection to [30]. Note that the combinatorics that should be satisfied by the I-equivariant sheaves (equipped with Frobenius endomorphisms) have been worked out in detail [ 62 , 65 ] [ 62 , 65 ] [62,65][62,65][62,65].

10.4. Tensor product decompositions

The tensor product decomposition of rational representations of G G GGG is deeply connected with our whole story due to the presentation (2.1). In [57], the geometry of flag varieties is used to deduce subtle information on the tensor products beyond the classical Littlewood-Richardson rule. It would be interesting to pursue their analogues in Q G rat Q G rat  Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat , possibly utilizing some modular interpretation [11] and connecting with the perspectives in [5].

10.5. The cotangent bundle of semi-infinite flags

A version of the cotangent bundle of Q G rat Q G rat  Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat  would make it possible to compare our results with the perspectives in [21,67,73]. In addition, its quantization should realize some numerics in Section 10.3. The author hopes to say a bit more on this in St. Petersburg.

ACKNOWLEDGMENTS

The works presented here could not be carried out without suggestions and interest by Misha Finkelberg. The author would like to express his deepest gratitude to him. The author also would like to thank Ivan Cherednik and Thomas Lam for sharing their insights over years, and Noriyuki Abe and Toshiyuki Tanisaki for their comments.

FUNDING

This work was partially supported by JSPS KAKENHI Grant Numbers JP26287004, JP19H01782, and JPJSBP120213210.

REFERENCES

[1] D. Anderson, L. Chen, H. H. Tseng, and H. Iritani, On the finiteness of quantum K K KKK-theory of a homogeneous space. 2018, arXiv:1804.04579v3.
[2] S. Arkhipov, A. Braverman, R. Bezrukavnikov, D. Gaitsgory, and I. Mirković, Modules over the small quantum group and semi-infinite flag manifold. Transform. Groups 10 (2005), no. 3-4, 279-362.
[3] S. Baldwin and S. Kumar, Positivity in T T TTT-equivariant K K KKK-theory of flag varieties associated to Kac-Moody groups. II. Represent. Theory 21 (2017), 35-60.
[4] J. Beck and H. Nakajima, Crystal bases and two-sided cells of quantum affine algebras. Duke Math. J. 123 (2004), no. 2, 335-402.
[5] P. Belkale, The tangent space to an enumerative problem. In Proceedings of the ICM 2010 (Hyderabad) II, pp. 405-426, Hindustan Book Agency, New Delhi, 2010 .
[6] R. Bezrukavnikov, M. Finkelberg, and I. Mirković, Equivariant homology and K K KKK-theory of affine Grassmannians and Toda lattices. Compos. Math. 141 (2005), no. 3, 746-768.
[7] A. Bouthier, Cohomologie étale des espaces d'arcs. 2015, arXiv:1509.02203v6.
[8] A. Braverman, Spaces of quasi-maps into the flag varieties and their applications. In Proceedings of the ICM 2006 (Madrid) II, pp. 1145-1170, Eur. Math. Soc., Zürich, 2006.
[9] A. Braverman and M. Finkelberg, Semi-infinite Schubert varieties and quantum K-theory of flag manifolds. J. Amer. Math. Soc. 27 (2014), no. 4, 1147-1168.
[10] A. Braverman and M. Finkelberg, Weyl modules and q q qqq-Whittaker functions. Math. Ann. 359 (2014), no. 1-2, 45-59.
[11] A. Braverman, M. Finkelberg, D. Gaitsgory, and I. Mirković, Intersection cohomology of Drinfeld's compactifications. Selecta Math. (N.S.) 8 (2002), no. 3, 381-418.
[12] M. Brion and S. Kumar, Frobenius splitting methods in geometry and representation theory. Progr. Math. 231, Birkhäuser Boston, Inc., Boston, MA, 2005.
[13] A. S. Buch, P.-E. Chaput, L. C. Mihalcea, and N. Perrin, Finiteness of cominuscule quantum K-theory. Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), no. 3, 477-494.
[14] V. Chari and B. Ion, BGG reciprocity for current algebras. Compos. Math. 151 (2015), no. 7, 1265-1287.
[15] I. Cherednik and S. Kato, Nonsymmetric Rogers-Ramanujan sums and thick Demazure modules. Adv. Math. 374 (2020), 107335, 57 p̀p.
[16] C. Chevalley, Sur les décompositions cellulaires des espaces G/B. In Algebraic groups and their generalizations: classical methods (University Park, PA, 1991), pp. 1-23, Amer. Math. Soc., Providence, RI, 1994.
[17] V. G. Drinfel'd, Quantum groups. In Proceedings of the ICM 1986 (Berkeley) Vol. 1, 2, pp. 798-820, Amer. Math. Soc., Providence, RI, 1987.
[18] B. Feigin, M. Finkelberg, A. Kuznetsov, and I. Mirković, Semi-infinite flags. II. Local and global intersection cohomology of quasimaps' spaces. In Differential topology, infinite-dimensional Lie algebras, and applications, pp. 113-148, Amer. Math. Soc. Transl. Ser. 2 194, Amer. Math. Soc., Providence, RI, 1999.
[19] B. L. Feйgin and E. V. Frenkel, Affine Kac-Moody algebras and semi-infinite flag manifolds. Comm. Math. Phys. 128 (1990), no. 1, 161-189.
[20] E. Feigin, S. Kato, and I. Makedonskyi, Representation theoretic realization of non-symmetric Macdonald polynomials at infinity. J. Reine Angew. Math. 764 (2020), 181-216.
[21] M. Finkelberg, Double affine Grassmannians and Coulomb branches of 3 d 3 d 3d3 d3d N = 4 N = 4 N=4\mathcal{N}=4N=4 quiver gauge theories. In Proceedings of the ICM 2018 (Rio de Janeiro), pp. 1283-1302, World Sci. Publ., Hackensack, NJ, 2018.
[22] M. Finkelberg and I. Mirković, Semi-infinite flags. I. Case of global curve P 1 P 1 P^(1)\mathbf{P}^{1}P1. In Differential topology, infinite-dimensional Lie algebras, and applications, pp. 81-112, Amer. Math. Soc. Transl. Ser. 2 194, Amer. Math. Soc., Providence, RI, 1999 .
[23] M. Finkelberg and A. Tsymbaliuk, Multiplicative slices, relativistic Toda and shifted quantum affine algebras. In Representations and nilpotent orbits of Lie algebraic systems, edited by M. Gorelik, V. Hinich, and A. Melnikov, pp. 133-304, Progr. Math. 330, Birkhäuser, Basel, 2019.
[24] S. Fishel, I. Grojnowski, and C. Teleman, The strong Macdonald conjecture and Hodge theory on the loop Grassmannian. Ann. of Math. (2) 168 (2008), no. 1, 175 220 175 − 220 175-220175-220175−220.
[25] G. Fourier and P. Littelmann, Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions. Adv. Math. 211 (2007), no. 2, 566-593.
[26] E. Frenkel and D. Gaitsgory, Localization of g g ggg-modules on the affine Grassmannian. Ann. of Math. (2) 170 (2009), no. 3, 1339-1381.
[27] E. Frenkel, D. Gaitsgory, D. Kazhdan, and K. Vilonen, Geometric realization of Whittaker functions and the Langlands conjecture. J. Amer. Math. Soc. 11 (1998), no. 2, 451-484.
[28] W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology. In Algebraic Geometry: Santa Cruz 1995, edited by R. L. J. Kollár and D. Morrison, pp. 45-96, Proc. Sympos. Pure Math. 62, Amer. Math. Soc., Providence, RI, 1995.
[29] W. Fulton and C. Woodward, On the quantum product of Schubert classes. J. Algebraic Geom. 13 (2004), 641-661.
[30] D. Gaitsgory, The semi-infinite intersection cohomology sheaf. Adv. Math. 327 (2018), 789-868.
[31] D. Gaitsgory, The local and global versions of the Whittaker category. Pure Appl. Math. Q. 16 (2020), no. 3, 775-904.
[32] A. Givental, Homological geometry and mirror symmetry. In Proceedings of the ICM 1994 (Zurich), pp. 472-480, Birkhäuser, Basel, 1995.
[33] A. Givental, On the WDVV equation in quantum K K KKK-theory. Michigan Math. J. 48 (2000), 295-304.
[34] A. Givental and Y. P. Lee, Quantum K K KKK-theory on flag manifolds, finite-difference Toda lattices and quantum groups. Invent. Math. 151 (2003), no. 1, 193-219.
[35] H. Iritani, T. Milanov, and V. Tonita, Reconstruction and convergence in quantum K K KKK-theory via difference equations. Int. Math. Res. Not. IMRN 2015 (2015), no. 11, 2887-2937.
[36] J. C. Jantzen, Representations of algebraic groups: second edition. Math. Surveys Monogr. 107, Amer. Math. Soc., Providence, RI, 2003.
[37] M. Jimbo, Solvable lattice models and quantum groups. In Proceedings of the ICM 1990 (Kyoto) I, II, pp. 1343-1352, Math. Soc. Japan, Tokyo, 1991.
[38] V. G. Kac, Infinite-dimensional Lie algebras. 3rd edn., Cambridge University Press, Cambridge, 1990.
[39] S. J. Kang and M. Kashiwara, Categorification of highest weight modules via Khovanov-Lauda-Rouquier algebras. Invent. Math. 190 (2012), no. 3, 699-742.
[40] M. Kashiwara, The flag manifold of Kac-Moody Lie algebra. In Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), pp. 161-190, Johns Hopkins Univ. Press, Baltimore, MD, 1989.
[41] M. Kashiwara, Crystallizing the q q qqq-analogue of universal enveloping algebras. In Proceedings of the ICM 1990 (Kyoto), pp. 791-797, Math. Soc. Japan, Tokyo, 1991.
[42] M. Kashiwara, Crystal bases of modified quantized enveloping algebra. Duke Math. J. 73 (1994), no. 2, 383-413.
[43] M. Kashiwara, Crystal bases and categorifications. In Proceedings of the ICM 2018 (Rio de Janeiro), pp. 249-258, World Sci. Publ., Hackensack, NJ, 2018.
[44] M. Kashiwara and T. Tanisaki, On Kazhdan-Lusztig conjectures. Sugaku Expositions 11 (1998), no. 2, 177-195.
[45] S. Kato, Darboux coordinates on the BFM spaces. 2020, arXiv:2008.01310v2.
[46] S. Kato, Demazure character formula for semi-infinite flag varieties. Math. Ann. 371 (2018), no. 3, 1769-1801.
[47] S. Kato, Loop structure on equivariant K K KKK-theory of semi-infinite flag manifolds. 2018, arXiv:1805.01718v6.
[48] S. Kato, On quantum K K KKK-groups of partial flag manifolds. 2019, arXiv:1906.09343v2.
[49] S. Kato, Frobenius splitting of thick flag manifolds of Kac-Moody algebras. Int. Math. Res. Not. IMRN 2020 (2020), no. 17, 5401-5427.
[50] S. Kato, Frobenius splitting of Schubert varieties of semi-infinite flag manifolds. Forum Math. Pi 9 (2021), e5, 56 pp.
[51] S. Kato and S. Loktev, A Weyl module stratification of integrable representations. Comm. Math. Phys. 368 (2019), 113-141.
[52] S. Kato, S. Naito, and D. Sagaki, Equivariant K K KKK-theory of semi-infinite flag manifolds and the Pieri-Chevalley formula. Duke Math. J. 169 (2020), no. 13, 2421-2500.
[53] M. Khovanov and A. D. Lauda, A diagrammatic approach to categorification of quantum groups. I. Represent. Theory 13 (2009), 309-347.
[54] M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry. Comm. Math. Phys. 164 (1994), no. 3, 525-562.
[55] B. Kostant and S. Kumar, T T TTT-equivariant K K KKK-theory of generalized flag varieties. J. Differential Geom. 32 (1990), no. 2, 549-603.
[56] S. Kumar, Kac-Moody groups, their flag varieties and representation theory. Progr. Math. 204, Birkhäuser Boston, Inc., Boston, MA, 2002.
[57] S. Kumar, Tensor product decomposition. In Proceedings of the ICM 2010 (Hyderabad) III, pp. 1226-1261, Hindustan Book Agency, New Delhi, 2010.
[58] T. Lam, C. Li, L. C. Mihalcea, and M. Shimozono, A conjectural Peterson isomorphism in K K KKK-theory. J. Algebra 513 (2018), 326-343.
[59] T. Lam and M. Shimozono, Quantum cohomology of G / P G / P G//PG / PG/P and homology of affine Grassmannian. Acta Math. 204 (2010), no. 1, 49-90.
[60] B. Leclerc, Cluster algebras and representation theory. In Proceedings of the ICM 2010 (Hyderabad) IV, pp. 2471-2488, Hindustan Book Agency, New Delhi, 2010.
[61] Y. P. Lee, Quantum K-theory. I. Foundations. Duke Math. J. 121 (2004), no. 3, 389-424.
[62] G. Lusztig, Hecke algebras and Jantzen's generic decomposition patterns. Adv. Math. 37 (1980), no. 1, 121-164.
[63] G. Lusztig, Intersection cohomology methods in representation theory. In Proceedings of the ICM 1990 (Kyoto) I, II, pp. 155-174, Math. Soc. Japan, Tokyo, 1991.
[64] G. Lusztig, Introduction to quantum groups. Progr. Math. 110, Birkhäuser, 1994.
[65] G. Lusztig, Bases in equivariant K K KKK-theory. Represent. Theory 2 (1998), 298-369.
[66] G. Lusztig, Study of a Z Z Z\mathbf{Z}Z-form of the coordinate ring of a reductive group. J. Amer Math. Soc. 22 (2009), no. 3, 739-769.
[67] D. Maulik and A. Okounkov, Quantum groups and quantum cohomology. Astérisque 408 (2019), ix+209 pp.
[68] V. B. Mehta and A. Ramanathan, Frobenius splitting and cohomology vanishing for Schubert varieties. Ann. of Math. (2) 122 (1985), no. 1, 27-40.
[69] L. C. Mihalcea, On equivariant quantum cohomology of homogeneous spaces: Chevalley formulae and algorithms. Duke Math. J. 140 (2007), no. 2, 321-350.
[70] I. Mirković and K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings. Ann. of Math. (2) 166 (2007), no. 1, 95 143 95 − 143 95-14395-14395−143.
[71] C. Mokler, An algebraic geometric model of an action of the face monoid associated to a Kac-Moody group on its building. J. Pure Appl. Algebra 219 (2017), 331-397.
[72] K. Naoi, Weyl modules, Demazure modules and finite crystals for non-simply laced type. Adv. Math. 229 (2012), no. 2, 875-934.
[73] A. Okounkov, On the crossroads of enumerative geometry and geometric representation theory. In Proceedings of the ICM 2018 (Rio de Janeiro), pp. 839-868, World Sci. Publ., Hackensack, NJ, 2018.
[74] D. Peterson, Quantum cohomology of G / P G / P G//PG / PG/P. Lecture at MIT, 1997.
[75] D. H. Peterson and V. G. Kac, Infinite flag varieties and conjugacy theorems. Proc. Natl. Acad. Sci. USA 80 (1983), 1778-1782.
[76] R. Rouquier, Quiver Hecke algebras and 2-Lie algebras. Algebra Colloq. 19 (2012), no. 2, 359-410.
[77] M. Varagnolo and E. Vasserot, Canonical bases and KLR-algebras. J. Reine Angew. Math. 659 (2011), 67-100.
[78] G. Williamson, Parity sheaves and the Hecke category. In Proceedings of the ICM 2018 (Rio de Janeiro), pp. 979-1015, World Sci. Publ., Hackensack, NJ, 2018.

SYU KATO

Department of Mathematics, Kyoto University, Oiwake Kita-Shirakawa, Sakyo, Kyoto 6068502, Japan, syuchan@math.kyoto-u.ac.jp

CHARACTER ESTIMATES FOR FINITE SIMPLE GROUPS AND APPLICATIONS

MICHAEL J. LARSEN

ABSTRACT

Let G G GGG be a finite simple group, χ χ chi\chiχ an irreducible complex character, and g g ggg an element of G G GGG. It is often desirable to have upper bounds for | χ ( g ) | | χ ( g ) | |chi(g)||\chi(g)||χ(g)| in terms of χ ( 1 ) χ ( 1 ) chi(1)\chi(1)χ(1) and some measure of the regularity of g g ggg. This paper reviews what is known in this direction and presents typical applications of such bounds: to proving certain products of conjugacy classes cover G G GGG, to solving word equations over G G GGG, and to counting homomorphisms from a Fuchsian group to G G GGG.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 20-02; Secondary 20C15, 20C30, 20C33, 20D06

1. INTRODUCTION

Let G G GGG be a finite group, χ χ chi\chiχ the character of an irreducible complex representation ρ ρ rho\rhoρ of G G GGG, and g g ggg an element of G G GGG. As the eigenvalues of ρ ( g ) ρ ( g ) rho(g)\rho(g)ρ(g) are roots of unity, the bound | χ ( g ) | χ ( 1 ) | χ ( g ) | ≤ χ ( 1 ) |chi(g)| <= chi(1)|\chi(g)| \leq \chi(1)|χ(g)|≤χ(1) is trivial. For central elements g g ggg, no stronger upper bound than χ ( 1 ) χ ( 1 ) chi(1)\chi(1)χ(1) is possible. However, according to Schur, we know that
g G χ ( g ) χ ( g ) ¯ = | G | ∑ g ∈ G   χ ( g ) χ ( g ) ¯ = | G | sum_(g in G)chi(g) bar(chi(g))=|G|\sum_{g \in G} \chi(g) \overline{\chi(g)}=|G|∑g∈Gχ(g)χ(g)¯=|G|
and since χ ( x ) = χ ( g ) χ ( x ) = χ ( g ) chi(x)=chi(g)\chi(x)=\chi(g)χ(x)=χ(g) for all x x xxx in the conjugacy class g G g G g^(G)g^{G}gG, we obtain the centralizer bound
| χ ( g ) | | G | | g G | = | C G ( g ) | | χ ( g ) | ≤ | G | g G = C G ( g ) |chi(g)| <= sqrt((|G|)/(|g^(G)|))=sqrt(|C_(G)(g)|)|\chi(g)| \leq \sqrt{\frac{|G|}{\left|g^{G}\right|}}=\sqrt{\left|C_{G}(g)\right|}|χ(g)|≤|G||gG|=|CG(g)|
Other known upper bounds typically hold only for special classes of groups.
This paper reviews what is known about character bounds when G G GGG is a finite simple group or is closely related to such a group. There is a substantial literature on upper bounds for character ratios | χ ( g ) | χ ( 1 ) | χ ( g ) | χ ( 1 ) (|chi(g)|)/(chi(1))\frac{|\chi(g)|}{\chi(1)}|χ(g)|χ(1); see Martin Liebeck's survey [29] for recent results and applications in the case of groups of Lie type. These bounds are typically weakest for characters χ χ chi\chiχ of low degree, which points to the desirability of exponential bounds, that is, bounds of the form | χ ( g ) | χ ( 1 ) α ( g ) | χ ( g ) | ≤ χ ( 1 ) α ( g ) |chi(g)| <= chi(1)^(alpha(g))|\chi(g)| \leq \chi(1)^{\alpha(g)}|χ(g)|≤χ(1)α(g), where the size of α ( g ) α ( g ) alpha(g)\alpha(g)α(g) is typically related to the size of the centralizer of g g ggg compared to | G | | G | |G||G||G|. The next two sections focus on alternating groups and groups of Lie type, respectively. The remaining sections give some applications of these results and present some open problems.

2. SYMMETRIC AND ALTERNATING GROUPS

Motivated by questions in probability theory, a number of people have considered character ratio bounds for symmetric groups. In this series of groups, unlike groups of Lie type, character ratios for nontrivial elements and nontrivial characters can be arbitrarily close to 1 . The worst case for G = S n G = S n G=S_(n)G=\mathrm{S}_{n}G=Sn is the ratio n 3 n 1 n − 3 n − 1 (n-3)/(n-1)\frac{n-3}{n-1}n−3n−1, achieved when g g ggg is a transposition and χ χ chi\chiχ is a character of degree n 1 n − 1 n-1n-1n−1. Persi Diaconis and Mehrdad Shahshahani considered the case that g g ggg is a transposition and χ χ chi\chiχ is any irreducible character, proving in [4] that if both the first row and the first column of the Young diagram for χ = χ λ χ = χ λ chi=chi_(lambda)\chi=\chi_{\lambda}χ=χλ have length n / 2 ≤ n / 2 <= n//2\leq n / 2≤n/2, then the character ratio is less than 1 / 2 1 / 2 1//21 / 21/2, while if, for instance, the first row satisfies λ 1 > n / 2 λ 1 > n / 2 lambda_(1) > n//2\lambda_{1}>n / 2λ1>n/2, then
0 < χ ( g ) χ ( 1 ) λ 1 ( λ 1 1 ) + ( n λ 1 1 ) ( n λ 1 2 ) 2 n ( n 1 ) 0 < χ ( g ) χ ( 1 ) ≤ λ 1 λ 1 − 1 + n − λ 1 − 1 n − λ 1 − 2 − 2 n ( n − 1 ) 0 < (chi(g))/(chi(1)) <= (lambda_(1)(lambda_(1)-1)+(n-lambda_(1)-1)(n-lambda_(1)-2)-2)/(n(n-1))0<\frac{\chi(g)}{\chi(1)} \leq \frac{\lambda_{1}\left(\lambda_{1}-1\right)+\left(n-\lambda_{1}-1\right)\left(n-\lambda_{1}-2\right)-2}{n(n-1)}0<χ(g)χ(1)≤λ1(λ1−1)+(n−λ1−1)(n−λ1−2)−2n(n−1)
A similar bound was given by Leopold Flatto, Andrew Odlyzko, and David Wales [8, тHEoREM 5.2].
Yuval Roichman [39] gave a character bound of the form
| χ ( g ) | χ ( 1 ) max ( λ 1 / n , λ 1 / n , c ) supp ( g ) | χ ( g ) | χ ( 1 ) ≤ max λ 1 / n , λ 1 ′ / n , c supp ⁡ ( g ) (|chi(g)|)/(chi(1)) <= max(lambda_(1)//n,lambda_(1)^(')//n,c)^(supp(g))\frac{|\chi(g)|}{\chi(1)} \leq \max \left(\lambda_{1} / n, \lambda_{1}^{\prime} / n, c\right)^{\operatorname{supp}(g)}|χ(g)|χ(1)≤max(λ1/n,λ1′/n,c)supp⁡(g)
where supp ( g ) supp ⁡ ( g ) supp(g)\operatorname{supp}(g)supp⁡(g) denotes the number of elements of { 1 , , n } { 1 , … , n } {1,dots,n}\{1, \ldots, n\}{1,…,n} not fixed by g g ggg, and c < 1 c < 1 c < 1c<1c<1 is an absolute constant. This reflects the fact that elements with high support tend to have
small centralizers. The bound is quite good when χ χ chi\chiχ has small degree. However, for large n n nnn, most characters of S n S n S_(n)\mathrm{S}_{n}Sn have degree greater than A n A n A^(n)A^{n}An for any fixed A A AAA, and for such characters, Roichman's bound is weaker than the centralizer bound for most elements g G g ∈ G g in Gg \in Gg∈G.
Philippe Biane [3] gave character ratio bounds for elements of bounded support and "balanced" characters, namely those where λ 1 / n λ 1 / n lambda_(1)//sqrtn\lambda_{1} / \sqrt{n}λ1/n and λ 1 / n λ 1 ′ / n lambda_(1)^(')//sqrtn\lambda_{1}^{\prime} / \sqrt{n}λ1′/n are bounded. By the work of Logan-Shepp [34] and VerÅ¡ik-Kerov [44], high degree characters are typically balanced. To be more precise, this is true for characters chosen randomly, weighted by the Plancherel measure. Amarpreet Rattan and Piotr Åšniady [38] generalized Biane's character bound so it applies whenever supp ( g ) supp ⁡ ( g ) supp(g)\operatorname{supp}(g)supp⁡(g) is small enough compared to n n nnn; if g g ggg cannot be expressed as the product of less than π Ï€ pi\piÏ€ transpositions, then
| χ ( g ) | χ ( 1 ) ( D max ( 1 , π 2 / n ) n ) π | χ ( g ) | χ ( 1 ) ≤ D max 1 , Ï€ 2 / n n Ï€ (|chi(g)|)/(chi(1)) <= ((D max(1,pi^(2)//n))/(sqrtn))^(pi)\frac{|\chi(g)|}{\chi(1)} \leq\left(\frac{D \max \left(1, \pi^{2} / n\right)}{\sqrt{n}}\right)^{\pi}|χ(g)|χ(1)≤(Dmax(1,Ï€2/n)n)Ï€
where D D DDD depends on the sizes of λ 1 / n λ 1 / n lambda_(1)//sqrtn\lambda_{1} / \sqrt{n}λ1/n and λ 1 / n λ 1 ′ / n lambda_(1)^(')//sqrtn\lambda_{1}^{\prime} / \sqrt{n}λ1′/n. Valentin Féray and Åšniady [7] proved a bound of the form
| χ ( g ) | χ ( 1 ) ( a max ( λ 1 , λ 1 , π ) n ) π | χ ( g ) | χ ( 1 ) ≤ a max λ 1 , λ 1 ′ , Ï€ n Ï€ (|chi(g)|)/(chi(1)) <= ((a max(lambda_(1),lambda_(1)^('),pi))/(n))^(pi)\frac{|\chi(g)|}{\chi(1)} \leq\left(\frac{a \max \left(\lambda_{1}, \lambda_{1}^{\prime}, \pi\right)}{n}\right)^{\pi}|χ(g)|χ(1)≤(amax(λ1,λ1′,Ï€)n)Ï€
which simultaneously improves on the results of [39] and [38].
Thomas Müller and Jan-Christoph Schlage-Puchta gave a character bound of exponential type [37, THEOREM B] which is good in a wide variety of situations. They proved that | χ ( g ) | χ ( 1 ) α ( g ) | χ ( g ) | ≤ χ ( 1 ) α ( g ) |chi(g)| <= chi(1)^(alpha(g))|\chi(g)| \leq \chi(1)^{\alpha(g)}|χ(g)|≤χ(1)α(g), where
α ( g ) = 1 ( ( 1 ( 1 / log n ) ) 1 12 log n log ( n / f i x ( g ) ) + 18 ) 1 α ( g ) = 1 − ( 1 − ( 1 / log ⁡ n ) ) − 1 12 log ⁡ n log ⁡ ( n / f i x ( g ) ) + 18 − 1 alpha(g)=1-((1-(1//log n))^(-1)(12 log n)/(log(n//fix(g)))+18)^(-1)\alpha(g)=1-\left((1-(1 / \log n))^{-1} \frac{12 \log n}{\log (n / \mathrm{fix}(g))}+18\right)^{-1}α(g)=1−((1−(1/log⁡n))−112log⁡nlog⁡(n/fix(g))+18)−1
Being exponential, it works well whether χ ( 1 ) χ ( 1 ) chi(1)\chi(1)χ(1) is large or small. The exponent is optimal, up to a multiplicative constant, for elements g g ggg consisting of many cycles, for instance, for involutions. However, it can be greatly improved upon for elements consisting of few cycles. In particular, α ( g ) α ( g ) alpha(g)\alpha(g)α(g) is no smaller when g g ggg is an n n nnn-cycle than when it is of shape 2 n / 2 2 n / 2 2^(n//2)2^{n / 2}2n/2.
Sergey Fomin and Nathan Lulov [9] gave a bound specifically for elements g g ggg of the shape r n / r r n / r r^(n//r)r^{n / r}rn/r. For fixed r r rrr and varying n n nnn, it takes the form
| χ ( g ) | = O ( n r 1 2 r χ ( 1 ) 1 / r ) | χ ( g ) | = O n r − 1 2 r χ ( 1 ) 1 / r |chi(g)|=O(n^((r-1)/(2r))chi(1)^(1//r))|\chi(g)|=O\left(n^{\frac{r-1}{2 r}} \chi(1)^{1 / r}\right)|χ(g)|=O(nr−12rχ(1)1/r)
so it is essentially a bound of exponential type. Aner Shalev and I gave an exponential bound [22] for elements g g ggg of arbitrary shape 1 a 1 2 a 2 1 a 1 2 a 2 … 1^(a_(1))2^(a_(2))dots1^{a_{1}} 2^{a_{2}} \ldots1a12a2… which is roughly comparable in strength to the Fomin-Lulov bound. Define the sequence e 1 , e 2 , e 1 , e 2 , … e_(1),e_(2),dotse_{1}, e_{2}, \ldotse1,e2,… such that for all k 1 k ≥ 1 k >= 1k \geq 1k≥1,
n e 1 + + e k = i = 1 k i a i n e 1 + ⋯ + e k = ∑ i = 1 k   i a i n^(e_(1)+cdots+e_(k))=sum_(i=1)^(k)ia_(i)n^{e_{1}+\cdots+e_{k}}=\sum_{i=1}^{k} i a_{i}ne1+⋯+ek=∑i=1kiai
Then
| χ ( g ) | χ ( 1 ) i = 1 n e i / i + o ( 1 ) | χ ( g ) | ≤ χ ( 1 ) ∑ i = 1 n   e i / i + o ( 1 ) |chi(g)| <= chi(1)^(sum_(i=1)^(n)e_(i)//i+o(1))|\chi(g)| \leq \chi(1)^{\sum_{i=1}^{n} e_{i} / i+o(1)}|χ(g)|≤χ(1)∑i=1nei/i+o(1)
This result is stronger than the exponential bound of Müller-Schlage-Puchta for almost all elements but inferior to it when the number of fixed points of g g ggg is very large.
None of these bounds can compete with the centralizer bound for elements consisting of very few cycles, for instance, for n n nnn-cycles, where the centralizer bound gives | χ ( g ) | n | χ ( g ) | ≤ n |chi(g)| <= sqrtn|\chi(g)| \leq \sqrt{n}|χ(g)|≤n. For such elements, the Murnaghan-Nakayama rule asserts | χ ( g ) | 1 | χ ( g ) | ≤ 1 |chi(g)| <= 1|\chi(g)| \leq 1|χ(g)|≤1, which is obviously optimal.
From symmetric group bounds, we easily obtain alternating group bounds of comparable strength. Recall that for λ λ λ ≠ λ ′ lambda!=lambda^(')\lambda \neq \lambda^{\prime}λ≠λ′, the characters χ λ χ λ chi_(lambda)\chi_{\lambda}χλ and χ λ χ λ ′ chi_(lambda^('))\chi_{\lambda^{\prime}}χλ′ restrict to the same irreducible character of A n A n A_(n)\mathrm{A}_{n}An. All other irreducible characters of A n A n A_(n)\mathrm{A}_{n}An arise from partitions satisfying λ = λ λ = λ ′ lambda=lambda^(')\lambda=\lambda^{\prime}λ=λ′; for each such λ λ lambda\lambdaλ, the restriction of χ λ χ λ chi_(lambda)\chi_{\lambda}χλ to A n A n A_(n)\mathrm{A}_{n}An decomposes as a sum of two irreducibles χ λ 1 χ λ 1 chi_(lambda)^(1)\chi_{\lambda}^{1}χλ1 and χ λ 2 χ λ 2 chi_(lambda)^(2)\chi_{\lambda}^{2}χλ2. The χ λ i χ λ i chi_(lambda)^(i)\chi_{\lambda}^{i}χλi take the character value χ λ ( g ) / 2 χ λ ( g ) / 2 chi_(lambda)(g)//2\chi_{\lambda}(g) / 2χλ(g)/2 for all g S n C g ∈ S n ∖ C g inS_(n)\\Cg \in \mathrm{S}_{n} \backslash Cg∈Sn∖C, where C C CCC is a single S n S n S_(n)\mathrm{S}_{n}Sn-conjugacy class which decomposes into two A n A n A_(n)\mathrm{A}_{n}An-conjugacy classes. For elements of C C CCC, a theorem of Frobenius gives character values, which are of the form
1 ± ± n 1 n k 2 1 ± ± n 1 ⋯ n k 2 (1+-sqrt(+-n_(1)cdotsn_(k)))/(2)\frac{1 \pm \sqrt{ \pm n_{1} \cdots n_{k}}}{2}1±±n1⋯nk2
where n i = λ i i n i = λ i − i n_(i)=lambda_(i)-in_{i}=\lambda_{i}-ini=λi−i for 1 i k 1 ≤ i ≤ k 1 <= i <= k1 \leq i \leq k1≤i≤k. Character degree estimates, like those in [22], now imply that | χ λ i ( g ) | χ λ ( 1 ) ε χ λ i ( g ) ≤ χ λ ( 1 ) ε |chi_(lambda)^(i)(g)| <= chi_(lambda)(1)^(epsi)\left|\chi_{\lambda}^{i}(g)\right| \leq \chi_{\lambda}(1)^{\varepsilon}|χλi(g)|≤χλ(1)ε whenever n n nnn is sufficiently large compared to ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0.

3. GROUPS OF LIE TYPE

Character estimates for finite simple groups of Lie type go back to the work of David Gluck [13-15]. Unlike in the case of alternating and symmetric groups, there is a uniform bound [15] on character ratios for nontrivial characters and nontrivial g g ggg, namely
| χ ( g ) | χ ( 1 ) 19 20 | χ ( g ) | χ ( 1 ) ≤ 19 20 (|chi(g)|)/(chi(1)) <= (19)/(20)\frac{|\chi(g)|}{\chi(1)} \leq \frac{19}{20}|χ(g)|χ(1)≤1920
When the cardinality q q qqq of the field of definition of G G GGG is large, this upper bound can be improved; Gluck [14] gives an upper bound of the form C / q C / q C//sqrtqC / \sqrt{q}C/q for large q q qqq. The q q qqq-exponent is optimal, since for odd q , PSL 2 ( q ) q , PSL 2 ⁡ ( q ) q,PSL_(2)(q)q, \operatorname{PSL}_{2}(q)q,PSL2⁡(q) has characters of degree q + 1 2 q + 1 2 (q+1)/(2)\frac{q+1}{2}q+12 or q 1 2 q − 1 2 (q-1)/(2)\frac{q-1}{2}q−12, and the value of such a character at a nontrivial unipotent element g g ggg is ± 1 ± ( 1 ) q 1 2 q 2 ± 1 ± ( − 1 ) q − 1 2 q 2 (+-1+-sqrt((-1)^((q-1)/(2))q))/(2)\frac{ \pm 1 \pm \sqrt{(-1)^{\frac{q-1}{2}} q}}{2}±1±(−1)q−12q2.
If G G GGG is a finite simple group of bounded rank, then χ ( 1 ) < | G | = O ( q D ) χ ( 1 ) < | G | = O q D chi(1) < |G|=O(q^(D))\chi(1)<|G|=O\left(q^{D}\right)χ(1)<|G|=O(qD), where D D DDD denotes the dimension associated to the Lie type of G G GGG. Therefore, the Gluck bound C / q C / q C//sqrtqC / \sqrt{q}C/q can be converted to an exponential bound | χ ( g ) | χ ( 1 ) α | χ ( g ) | ≤ χ ( 1 ) α |chi(g)| <= chi(1)^(alpha)|\chi(g)| \leq \chi(1)^{\alpha}|χ(g)|≤χ(1)α, where α < 1 α < 1 alpha < 1\alpha<1α<1 depends only on the rank. To achieve exponential bounds in general, therefore, it suffices to limit our attention to the case that G G GGG is a classical group, that is, one of PSL r + 1 ( q ) , PSU r + 1 ( q ) , P Ω ± 2 r ( q ) PSL r + 1 ⁡ ( q ) , PSU r + 1 ⁡ ( q ) , P Ω ± 2 r ( q ) PSL_(r+1)(q),PSU_(r+1)(q),POmega(+-)/(2r)(q)\operatorname{PSL}_{r+1}(q), \operatorname{PSU}_{r+1}(q), \mathrm{P} \Omega \frac{ \pm}{2 r}(q)PSLr+1⁡(q),PSUr+1⁡(q),PΩ±2r(q), P S p 2 r ( q ) P S p 2 r ( q ) PSp_(2r)(q)\mathrm{PSp}_{2 r}(q)PSp2r(q), or P Ω 2 r + 1 ( q ) P Ω 2 r + 1 ( q ) POmega_(2r+1)(q)\mathrm{P} \Omega_{2 r+1}(q)PΩ2r+1(q).
We cannot expect that character ratios go to 0 as the order of a classical group goes to infinity. For instance, let G = P S L r + 1 ( q ) G = P S L r + 1 ( q ) G=PSL_(r+1)(q)G=\mathrm{PSL}_{r+1}(q)G=PSLr+1(q). The permutation representation associated with the action of G G GGG on P F q r P F q r PF_(q)^(r)\mathbb{P} F_{q}^{r}PFqr can be expressed as χ + 1 χ + 1 chi+1\chi+1χ+1, for χ χ chi\chiχ irreducible. Let g g ggg be the image of a transvection in SL r + 1 ( F q ) SL r + 1 ⁡ F q SL_(r+1)(F_(q))\operatorname{SL}_{r+1}\left(\mathbb{F}_{q}\right)SLr+1⁡(Fq) in G G GGG. Then the fixed points of g g ggg form a hyperplane in P P q n P P q n PP_(q)^(n)\mathbb{P P}_{q}^{n}PPqn, and χ ( g ) = q n 1 + q n 2 + + q χ ( g ) = q n − 1 + q n − 2 + ⋯ + q chi(g)=q^(n-1)+q^(n-2)+cdots+q\chi(g)=q^{n-1}+q^{n-2}+\cdots+qχ(g)=qn−1+qn−2+⋯+q. Thus,
lim n χ ( g ) χ ( 1 ) = 1 q lim n → ∞   χ ( g ) χ ( 1 ) = 1 q lim_(n rarr oo)(chi(g))/(chi(1))=(1)/(q)\lim _{n \rightarrow \infty} \frac{\chi(g)}{\chi(1)}=\frac{1}{q}limn→∞χ(g)χ(1)=1q
Defining the support supp ( g ) supp ⁡ ( g ) supp(g)\operatorname{supp}(g)supp⁡(g) as the smallest codimension of any eigenspace of g g ggg for the natural projective representation of G G GGG, the elements g g ggg in the above example have constant support 1 even as the rank of G G GGG goes to infinity. Shalev, Pham Huu Tiep, and I proved [24, THEOREM 4.3.6] that as the support goes to infinity, the character ratio goes to 0 :
| χ ( g ) | χ ( 1 ) q supp ( g ) / 481 | χ ( g ) | χ ( 1 ) ≤ q − supp ⁡ ( g ) / 481 (|chi(g)|)/(chi(1)) <= q^(-sqrt(supp(g))//481)\frac{|\chi(g)|}{\chi(1)} \leq q^{-\sqrt{\operatorname{supp}(g)} / 481}|χ(g)|χ(1)≤q−supp⁡(g)/481
This falls well short of a uniform exponential character bound, even for elements of maximal support. Robert Guralnick, Tiep, and I found uniform exponential bounds for elements g g ggg whose centralizer is small compared to the order of G G GGG. For instance, we proved [16, THEOREM 1.4] that if G G GGG is of the form PSL n ( q ) PSL n ⁡ ( q ) PSL_(n)(q)\operatorname{PSL}_{n}(q)PSLn⁡(q) or PSU n ( q ) PSU n ⁡ ( q ) PSU_(n)(q)\operatorname{PSU}_{n}(q)PSUn⁡(q) and | C G ( g ) | q n 2 / 12 C G ( g ) ≤ q n 2 / 12 |C_(G)(g)| <= q^(n^(2)//12)\left|C_{G}(g)\right| \leq q^{n^{2} / 12}|CG(g)|≤qn2/12, then | χ ( g ) | χ ( 1 ) 8 / 9 | χ ( g ) | ≤ χ ( 1 ) 8 / 9 |chi(g)| <= chi(1)^(8//9)|\chi(g)| \leq \chi(1)^{8 / 9}|χ(g)|≤χ(1)8/9. More generally, but less explicitly, we proved [17, THEOREM 1.3] that for all ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0, there exists δ > 0 δ > 0 delta > 0\delta>0δ>0 such that | C G ( g ) | | G | δ C G ( g ) ≤ | G | δ |C_(G)(g)| <= |G|^(delta)\left|C_{G}(g)\right| \leq|G|^{\delta}|CG(g)|≤|G|δ implies | χ ( g ) | χ ( 1 ) ε | χ ( g ) | ≤ χ ( 1 ) ε |chi(g)| <= chi(1)^(epsi)|\chi(g)| \leq \chi(1)^{\varepsilon}|χ(g)|≤χ(1)ε. However, the method of these papers applies only to elements with small centralizer, for instance, it does not give any bound at all for involutions.
This defect was remedied in the sequel [28], which proved that for all positive δ < 1 δ < 1 delta < 1\delta<1δ<1 there exists ε < 1 ε < 1 epsi < 1\varepsilon<1ε<1 such that | C G ( g ) | | G | δ C G ( g ) ≤ | G | δ |C_(G)(g)| <= |G|^(delta)\left|C_{G}(g)\right| \leq|G|^{\delta}|CG(g)|≤|G|δ implies | χ ( g ) | χ ( 1 ) ε | χ ( g ) | ≤ χ ( 1 ) ε |chi(g)| <= chi(1)^(epsi)|\chi(g)| \leq \chi(1)^{\varepsilon}|χ(g)|≤χ(1)ε. More precisely, | χ ( g ) | | χ ( g ) | ≤ |chi(g)| <=|\chi(g)| \leq|χ(g)|≤ χ ( 1 ) α ( g ) χ ( 1 ) α ( g ) chi(1)^(alpha(g))\chi(1)^{\alpha(g)}χ(1)α(g) where
α ( g ) = 1 c + c log | C G ( g ) | log | G | α ( g ) = 1 − c + c log ⁡ C G ( g ) log ⁡ | G | alpha(g)=1-c+c(log|C_(G)(g)|)/(log |G|)\alpha(g)=1-c+c \frac{\log \left|C_{G}(g)\right|}{\log |G|}α(g)=1−c+clog⁡|CG(g)|log⁡|G|
and c > 0 c > 0 c > 0c>0c>0 is an absolute constant, which can be made explicit (but is, unfortunately, extremely small). This theorem holds more generally for quasisimple finite groups of Lie type.
For many elements g g ggg in a classical group of rank r r rrr, much better exponents are available, thanks to the work of Roman Bezrukavnikov, Liebeck, Shalev, and Tiep [2]. For q q qqq odd, if the centralizer of g g ggg is a proper split Levi subgroup, then | χ ( g ) | f ( r ) χ ( 1 ) α ( g ) | χ ( g ) | ≤ f ( r ) χ ( 1 ) α ( g ) |chi(g)| <= f(r)chi(1)^(alpha(g))|\chi(g)| \leq f(r) \chi(1)^{\alpha(g)}|χ(g)|≤f(r)χ(1)α(g), where α ( g ) α ( g ) alpha(g)\alpha(g)α(g) is an explicitly computable rational number which is known to be optimal in many cases. This idea was further developed by Jay Taylor and Tiep, who proved [43], among other things, that for every nontrivial element g PSL n ( q ) g ∈ PSL n ⁡ ( q ) g inPSL_(n)(q)g \in \operatorname{PSL}_{n}(q)g∈PSLn⁡(q),
| χ ( g ) | h ( r ) χ ( 1 ) n 1 n 2 | χ ( g ) | ≤ h ( r ) χ ( 1 ) n − 1 n − 2 |chi(g)| <= h(r)chi(1)^((n-1)/(n-2))|\chi(g)| \leq h(r) \chi(1)^{\frac{n-1}{n-2}}|χ(g)|≤h(r)χ(1)n−1n−2
All of these estimates are poor for elements with small centralizers, such as regular elements. A general result, due to Shelly Garion, Alexander Lubotzky, and myself, which sometimes gives reasonably good bounds for regular elements, is the following [10, THEOREM 3]. Let G G GGG be a finite group, not necessarily simple, and g g ggg an element of G G GGG whose centralizer A A AAA is abelian. Suppose A 1 , , A n A 1 , … , A n A_(1),dots,A_(n)A_{1}, \ldots, A_{n}A1,…,An are subgroups of A A AAA not containing g g ggg such that the centralizer of every element of A i A i A ∖ ⋃ i   A i A\\uuu_(i)A_(i)A \backslash \bigcup_{i} A_{i}A∖⋃iAi is A A AAA. Then, for every irreducible character of G G GGG,
| χ ( g ) | ( 4 / 3 ) n [ N G ( A ) : A ] | χ ( g ) | ≤ ( 4 / 3 ) n N G ( A ) : A |chi(g)| <= (4//sqrt3)^(n)[N_(G)(A):A]|\chi(g)| \leq(4 / \sqrt{3})^{n}\left[N_{G}(A): A\right]|χ(g)|≤(4/3)n[NG(A):A]
For example, this gives an upper bound of 2 ( n 1 ) 2 / 3 2 ( n − 1 ) 2 / 3 2(n-1)^(2)//sqrt32(n-1)^{2} / \sqrt{3}2(n−1)2/3 for | χ ( g ) | | χ ( g ) | |chi(g)||\chi(g)||χ(g)| when G = PSL n ( q ) G = PSL n ⁡ ( q ) G=PSL_(n)(q)G=\operatorname{PSL}_{n}(q)G=PSLn⁡(q) and g g ggg is the image of an element with irreducible characteristic polynomial. It would be nice to have optimal upper bounds for | χ ( g ) | | χ ( g ) | |chi(g)||\chi(g)||χ(g)| for general regular semisimple elements g g ggg.

4. PRODUCTS OF CONJUGACY CLASSES

If C 1 , , C n C 1 , … , C n C_(1),dots,C_(n)C_{1}, \ldots, C_{n}C1,…,Cn are conjugacy classes of a finite group G G GGG, then the number N N NNN of n n nnn tuples ( g 1 , , g n ) C 1 × × C n g 1 , … , g n ∈ C 1 × ⋯ × C n (g_(1),dots,g_(n))inC_(1)xx cdots xxC_(n)\left(g_{1}, \ldots, g_{n}\right) \in C_{1} \times \cdots \times C_{n}(g1,…,gn)∈C1×⋯×Cn satisfying g 1 g 2 g n = 1 g 1 g 2 ⋯ g n = 1 g_(1)g_(2)cdotsg_(n)=1g_{1} g_{2} \cdots g_{n}=1g1g2⋯gn=1 is given by the Frobenius formula
N = | C 1 | | C n | | G | χ χ ( C 1 ) χ ( C n ) χ ( 1 ) n 2 N = C 1 ⋯ C n | G | ∑ χ   χ C 1 ⋯ χ C n χ ( 1 ) n − 2 N=(|C_(1)|cdots|C_(n)|)/(|G|)sum_(chi)(chi(C_(1))cdots chi(C_(n)))/(chi(1)^(n-2))N=\frac{\left|C_{1}\right| \cdots\left|C_{n}\right|}{|G|} \sum_{\chi} \frac{\chi\left(C_{1}\right) \cdots \chi\left(C_{n}\right)}{\chi(1)^{n-2}}N=|C1|⋯|Cn||G|∑χχ(C1)⋯χ(Cn)χ(1)n−2
where χ χ chi\chiχ ranges over all irreducible characters of G G GGG. In conjunction with upper bounds for the | χ ( C i ) | χ C i |chi(C_(i))|\left|\chi\left(C_{i}\right)\right||χ(Ci)|, this can sometimes be used to prove that N 0 N ≠ 0 N!=0N \neq 0N≠0, as the contribution from χ = 1 χ = 1 chi=1\chi=1χ=1 often dominates the sum. Exponential bounds for the χ ( C i ) χ C i chi(C_(i))\chi\left(C_{i}\right)χ(Ci) are especially convenient, since results of Liebeck and Shalev [32] give a great deal of information about when we can expect
χ 1 χ ( 1 ) s < 1 ∑ χ ≠ 1   χ ( 1 ) − s < 1 sum_(chi!=1)chi(1)^(-s) < 1\sum_{\chi \neq 1} \chi(1)^{-s}<1∑χ≠1χ(1)−s<1
A well-known conjecture attributed to Thompson asserts that for every finite simple group G G GGG, there exists a conjugacy class C C CCC such that C 2 = G C 2 = G C^(2)=GC^{2}=GC2=G. Thanks to work of Erich Ellers and Nikolai Gordeev [6], we know that this is true except for a list of possible counterexamples, all finite simple groups of Lie type with q 8 q ≤ 8 q <= 8q \leq 8q≤8. Tiep and I used our uniform exponential bounds to show that several of the infinite families on this list, in particular, the symplectic groups for all q 2 q ≥ 2 q >= 2q \geq 2q≥2, can be eliminated in sufficiently high rank [28, тHEoREM 7.7]. It would be interesting if these results could be extended to the remaining families on the list, giving an asymptotic version of Thompson's conjecture.
Andrew Gleason and Cheng-hao Xu [18,19] proved Thompson's conjecture for alternating groups, using the conjugacy class of an n n nnn-cycle if n n nnn is odd or a permutation of shape 2 1 ( n 2 ) 1 2 1 ( n − 2 ) 1 2^(1)(n-2)^(1)2^{1}(n-2)^{1}21(n−2)1 if n n nnn is even. In [22, THEOREM 1.13], Shalev and I proved that in the limit n n → ∞ n rarr oon \rightarrow \inftyn→∞ the probability that a randomly chosen g A n g ∈ A n g inA_(n)g \in \mathrm{A}_{n}g∈An belongs to a conjugacy class with C 2 = A n C 2 = A n C^(2)=A_(n)C^{2}=\mathrm{A}_{n}C2=An rapidly approaches 1 .
The analogous claim cannot be true for all finite simple groups since C 2 = G C 2 = G C^(2)=GC^{2}=GC2=G implies that C = C 1 C = C − 1 C=C^(-1)C=C^{-1}C=C−1, and for, e.g., PSL 3 ( q ) PSL 3 ⁡ ( q ) PSL_(3)(q)\operatorname{PSL}_{3}(q)PSL3⁡(q) as q q → ∞ q rarr ooq \rightarrow \inftyq→∞, the probability that a random element is real goes to 0 . However, there are several variants of this question which do not have an obvious counterexample. As the order of G G GGG tends to infinity, does the probability that a random real element belongs to a conjugacy class with C 2 = G C 2 = G C^(2)=GC^{2}=GC2=G approach 1? Does the probability that a random element g g ggg belongs to a conjugacy class C C CCC with C 2 { 1 } = G C 2 ∪ { 1 } = G C^(2)uu{1}=GC^{2} \cup\{1\}=GC2∪{1}=G approach 1? Also, as the order of G G GGG tends to infinity, does the probability that a random element belongs to a conjugacy class with C C 1 = G C C − 1 = G CC^(-1)=GC C^{-1}=GCC−1=G approach 1?
The weaker claim that every element g G g ∈ G g in Gg \in Gg∈G lies in C C 1 C C − 1 CC^(-1)C C^{-1}CC−1 for some conjugacy class (depending, perhaps, on g g ggg ) is equivalent to the statement that every element of G G GGG is a commutator. This was was an old conjecture of Ore and is now a theorem of Liebeck, Eamonn O'Brien, Shalev, and Tiep [30].
One can also ask about S 2 S 2 S^(2)S^{2}S2 where S S SSS is an arbitrary conjugation-invariant subset of G G GGG. On naive probabilistic grounds, it might seem plausible that given ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0 fixed, for G G GGG sufficiently large, every normal subset of G G GGG with at least ε | G | ε | G | epsi|G|\varepsilon|G|ε|G| elements satisfies S 2 = G S 2 = G S^(2)=GS^{2}=GS2=G. However, a moment's reflection shows that, unless ε > 1 2 ε > 1 2 epsi > (1)/(2)\varepsilon>\frac{1}{2}ε>12, there is no reason to expect 1 S 2 1 ∈ S 2 1inS^(2)1 \in S^{2}1∈S2.
Is it true, for G G GGG sufficiently large, that S 2 { 1 } = G S 2 ∪ { 1 } = G S^(2)uu{1}=GS^{2} \cup\{1\}=GS2∪{1}=G ? For alternating groups and for groups of Lie type in bounded rank, the answer is affirmative [26], but we do not know in general.
In a different direction, given a conjugacy class C C CCC, how large must n n nnn be so that the n n nnnth power C n C n C^(n)C^{n}Cn is all of G G GGG ? More generally, given conjugacy classes C 1 , , C n C 1 , … , C n C_(1),dots,C_(n)C_{1}, \ldots, C_{n}C1,…,Cn with sufficiently strong character bounds, the Frobenius formula can be used to show that each element of G G GGG is represented as a product g 1 g n g 1 ⋯ g n g_(1)cdotsg_(n)g_{1} \cdots g_{n}g1⋯gn, with g i C i g i ∈ C i g_(i)inC_(i)g_{i} \in C_{i}gi∈Ci, in approximately | C 1 | | C n | | G | C 1 ⋯ C n | G | (|C_(1)|cdots|C_(n)|)/(|G|)\frac{\left|C_{1}\right| \cdots\left|C_{n}\right|}{|G|}|C1|⋯|Cn||G| ways. For instance, it follows from the exponential character bounds given above that there exists an absolute constant k k kkk such that if G G GGG is a finite simple group of Lie type and C 1 , , C n C 1 , … , C n C_(1),dots,C_(n)C_{1}, \ldots, C_{n}C1,…,Cn are conjugacy classes in G G GGG satisfying | C 1 | | C n | > | G | k C 1 ⋯ C n > | G | k |C_(1)|cdots|C_(n)| > |G|^(k)\left|C_{1}\right| \cdots\left|C_{n}\right|>|G|^{k}|C1|⋯|Cn|>|G|k, then for each g G g ∈ G g in Gg \in Gg∈G,
| { ( g 1 , , g n ) C 1 × × C n g 1 g n = g } | = ( 1 + o ( 1 ) ) | C 1 | | C n | | G | g 1 , … , g n ∈ C 1 × ⋯ × C n ∣ g 1 ⋯ g n = g = ( 1 + o ( 1 ) ) C 1 ⋯ C n | G | |{(g_(1),dots,g_(n))inC_(1)xx cdots xxC_(n)∣g_(1)cdotsg_(n)=g}|=(1+o(1))(|C_(1)|cdots|C_(n)|)/(|G|)\left|\left\{\left(g_{1}, \ldots, g_{n}\right) \in C_{1} \times \cdots \times C_{n} \mid g_{1} \cdots g_{n}=g\right\}\right|=(1+o(1)) \frac{\left|C_{1}\right| \cdots\left|C_{n}\right|}{|G|}|{(g1,…,gn)∈C1×⋯×Cn∣g1⋯gn=g}|=(1+o(1))|C1|⋯|Cn||G|
Via Lang-Weil estimates, this further implies that if C _ 1 , , C _ n C _ 1 , … , C _ n C__(1),dots,C__(n)\underline{C}_{1}, \ldots, \underline{C}_{n}C_1,…,C_n are conjugacy classes of a simple algebraic group G _ G _ G_\underline{G}G_, and
dim C _ 1 + + dim C _ n > k dim G _ dim ⁡ C _ 1 + ⋯ + dim ⁡ C _ n > k dim ⁡ G _ dim C__(1)+cdots+dim C__(n) > k dim G_\operatorname{dim} \underline{C}_{1}+\cdots+\operatorname{dim} \underline{C}_{n}>k \operatorname{dim} \underline{G}dim⁡C_1+⋯+dim⁡C_n>kdim⁡G_
then the product morphism of varieties C _ 1 × × C _ n G _ C _ 1 × ⋯ × C _ n → G _ C__(1)xx cdots xxC__(n)rarrG_\underline{C}_{1} \times \cdots \times \underline{C}_{n} \rightarrow \underline{G}C_1×⋯×C_n→G_ has the property that every fiber is of dimension dim C _ 1 + + dim C _ n dim G _ dim ⁡ C _ 1 + ⋯ + dim ⁡ C _ n − dim ⁡ G _ dim C__(1)+cdots+dim C__(n)-dim G_\operatorname{dim} \underline{C}_{1}+\cdots+\operatorname{dim} \underline{C}_{n}-\operatorname{dim} \underline{G}dim⁡C_1+⋯+dim⁡C_n−dim⁡G_.
In the special case that C 1 = = C n = C C 1 = ⋯ = C n = C C_(1)=cdots=C_(n)=CC_{1}=\cdots=C_{n}=CC1=⋯=Cn=C, the question of the distribution of products g 1 g n , g i C g 1 ⋯ g n , g i ∈ C g_(1)cdotsg_(n),g_(i)in Cg_{1} \cdots g_{n}, g_{i} \in Cg1⋯gn,gi∈C, can be expressed in terms of the mixing time of the random walk on the Cayley graph of ( G , C G , C G,CG, CG,C ). A consequence of the exponential character bounds [28] is that for groups of Lie type, the mixing time of such a random walk is O ( log | G | / log | C | ) O ( log ⁡ | G | / log ⁡ | C | ) O(log |G|//log |C|)O(\log |G| / \log |C|)O(log⁡|G|/log⁡|C|). This is the same order of growth as the diameter of the Cayley graph, thus settling conjectures of Lubotzky [35, P. 179] and Shalev [42, CONJECTURE 4.3].
The situation is different for alternating groups G = A n G = A n G=A_(n)G=\mathrm{A}_{n}G=An. For instance, if C C CCC is the class of 3-cycles and n 6 n ≥ 6 n >= 6n \geq 6n≥6, then log | G | / log | C | < n log ⁡ | G | / log ⁡ | C | < n log |G|//log |C| < n\log |G| / \log |C|<nlog⁡|G|/log⁡|C|<n, and C n / 2 = G C ⌊ n / 2 ⌋ = G C^(|__ n//2__|)=GC^{\lfloor n / 2\rfloor}=GC⌊n/2⌋=G [5, THEOREM 9.1]. However, for any fixed k k kkk, the probability that the product of k n k n knk nkn random 3 -cycles g i g i g_(i)g_{i}gi fixes 1 is at least the probability that each individual g i g i g_(i)g_{i}gi fixes 1 , which goes to e 3 k e − 3 k e^(-3k)e^{-3 k}e−3k as n n → ∞ n rarr oon \rightarrow \inftyn→∞. Thus the expected number of fixed points of g 1 g n g 1 ⋯ g n g_(1)cdotsg_(n)g_{1} \cdots g_{n}g1⋯gn grows linearly with n n nnn. It would be interesting to know, for general C A n C ⊂ A n C subA_(n)C \subset \mathrm{A}_{n}C⊂An, what the mixing time is.

5. WARING'S PROBLEM

Waring's problem for finite simple groups originally meant the following question. Does there exist a function f : N N f : N → N f:NrarrNf: \mathbb{N} \rightarrow \mathbb{N}f:N→N such that for all positive integers n n nnn and all sufficiently large finite simple groups G G GGG (in terms of n n nnn ), every element of G G GGG is a product of f ( n ) n f ( n ) n f(n)nf(n) nf(n)nth powers? Positive solutions were given by Martinez-Zelmanov [36] and Saxl-Wilson [40].
This can be extended as follows. Let w w www denote a nontrivial element in any free group F d F d F_(d)F_{d}Fd. For every finite simple group G , w G , w G,wG, wG,w determines a function G d G G d → G G^(d)rarr GG^{d} \rightarrow GGd→G. We replace the n n nnnth powers with word values, that is, elements of G G GGG in the image of w w www. Liebeck and Shalev proved [31] that for G G GGG sufficiently large (in terms of w w www ), every element of G G GGG can be written as a product of a bounded number of word values (where the bound may depend on w w www, just as
in the classical version of Waring's problem, the minimum number of the n n nnnth powers needed to represent a given integer may depend on n n nnn ).
It was therefore, perhaps, surprising when Shalev proved [41] that the Waring number for finite simple groups is uniform in w w www and is, in fact, at most three. This has now been improved to the optimal bound, two [23,24]. More generally, for any two nontrivial words w 1 w 1 w_(1)w_{1}w1 and w 2 w 2 w_(2)w_{2}w2, if G G GGG is a sufficiently large finite simple group, every element of G G GGG is a product of their word values. In fact, it is even possible [27] to choose subsets S 1 S 1 S_(1)S_{1}S1 and S 2 S 2 S_(2)S_{2}S2 of the sets of word values of w 1 w 1 w_(1)w_{1}w1 and w 2 w 2 w_(2)w_{2}w2 such that S 1 S 2 = G S 1 S 2 = G S_(1)S_(2)=GS_{1} S_{2}=GS1S2=G and | S i | = O ( | G | 1 / 2 log 1 / 2 | G | ) S i = O | G | 1 / 2 log 1 / 2 ⁡ | G | |S_(i)|=O(|G|^(1//2)log^(1//2)|G|)\left|S_{i}\right|=O\left(|G|^{1 / 2} \log ^{1 / 2}|G|\right)|Si|=O(|G|1/2log1/2⁡|G|). The set of values of any word is a union of conjugacy classes, and the basic strategy of the proof is to try to find conjugacy classes C 1 C 1 C_(1)C_{1}C1 and C 2 C 2 C_(2)C_{2}C2 contained in the word values of w 1 w 1 w_(1)w_{1}w1 and w 2 w 2 w_(2)w_{2}w2, respectively, such that C 1 C 2 = G C 1 C 2 = G C_(1)C_(2)=GC_{1} C_{2}=GC1C2=G and very few elements of G G GGG have significantly fewer representations as such products than one would expect. Then a random choice of subsets S i C i S i ⊂ C i S_(i)subC_(i)S_{i} \subset C_{i}Si⊂Ci of suitable size can almost always be slightly modified to work.
In general, the probability distribution on the word values of w w www obtained by evaluation at a uniformly distributed random element of G d G d G^(d)G^{d}Gd is far from uniform. For instance, for g A 3 n g ∈ A 3 n g inA_(3n)g \in \mathrm{A}_{3 n}g∈A3n uniformly distributed, the probability that g 3 = 1 g 3 = 1 g^(3)=1g^{3}=1g3=1 is at least | A 3 n | 1 A 3 n − 1 |A_(3n)|^(-1)\left|\mathrm{A}_{3 n}\right|^{-1}|A3n|−1 times the number of elements of shape 3 n 3 n 3^(n)3^{n}3n, i.e.,
( 3 n 1 ) ( 3 n 2 ) ( 3 n 4 ) ( 3 n 5 ) ( 2 ) ( 1 ) > ( 3 n 1 ) ! 2 3 > | A 3 n | 2 3 1 3 n ( 3 n − 1 ) ( 3 n − 2 ) ⋅ ( 3 n − 4 ) ( 3 n − 5 ) ⋯ ( 2 ) ( 1 ) > ( 3 n − 1 ) ! 2 3 > A 3 n 2 3 − 1 3 n (3n-1)(3n-2)*(3n-4)(3n-5)cdots(2)(1) > (3n-1)!^((2)/(3)) > |A_(3n)|^((2)/(3)-(1)/(3n))(3 n-1)(3 n-2) \cdot(3 n-4)(3 n-5) \cdots(2)(1)>(3 n-1)!^{\frac{2}{3}}>\left|A_{3 n}\right|^{\frac{2}{3}-\frac{1}{3 n}}(3n−1)(3n−2)⋅(3n−4)(3n−5)⋯(2)(1)>(3n−1)!23>|A3n|23−13n
for n n nnn sufficiently large. Thus, setting w 1 = w 2 = x 3 w 1 = w 2 = x 3 w_(1)=w_(2)=x^(3)w_{1}=w_{2}=x^{3}w1=w2=x3, the probability that the product of cubes of two randomly chosen elements is 1 is at least | A 3 n | 2 / 3 2 / 3 n A 3 n − 2 / 3 − 2 / 3 n |A_(3n)|^(-2//3-2//3n)\left|A_{3 n}\right|^{-2 / 3-2 / 3 n}|A3n|−2/3−2/3n, which, for large n n nnn, makes the distribution far from uniform, at least in the L L ∞ L^(oo)L^{\infty}L∞ sense.
Using exponential character estimates, Shalev, Tiep, and I proved [25, theorem 4] that for any word w w www, there exists k k kkk such that as | G | | G | → ∞ |G|rarr oo|G| \rightarrow \infty|G|→∞, the L L ∞ L^(oo)L^{\infty}L∞-deviation from uniformity in the product of k k kkk independent randomly generated values of w w www goes to 0 . The dependence of k k kkk on w w www is unavoidable, as the above example suggests. On the other hand, the L 1 L 1 L^(1)L^{1}L1-deviation from uniformity goes to 0 in the product of two independent randomly generated values of w w www, for any nontrivial word w w www [25, THEOREM 1]. I do not know what to expect for L p L p L^(p)L^{p}Lp-deviation for 1 < p < 1 < p < ∞ 1 < p < oo1<p<\infty1<p<∞.

6. FUCHSIAN GROUPS

For g , m 0 g , m ≥ 0 g,m >= 0g, m \geq 0g,m≥0, let d 1 , , d m 2 d 1 , … , d m ≥ 2 d_(1),dots,d_(m) >= 2d_{1}, \ldots, d_{m} \geq 2d1,…,dm≥2 be integers. For
Γ = x 1 , , x m , y 1 , , y g , z 1 , , z g | x 1 d 1 , , x m d m x 1 x m [ y 1 , z 1 ] [ y g , z g ] Γ = x 1 , … , x m , y 1 , … , y g , z 1 , … , z g x 1 d 1 , … , x m d m x 1 ⋯ x m y 1 , z 1 ⋯ y g , z g {:[Gamma=(:x_(1),dots,x_(m),y_(1),dots,y_(g),z_(1),dots,z_(g)|x_(1)^(d_(1))","dots","x_(m)^(d_(m))],[{:x_(1)cdotsx_(m)[y_(1),z_(1)]cdots[y_(g),z_(g)]:)]:}\begin{aligned} \Gamma=\left\langle x_{1}, \ldots, x_{m}, y_{1}, \ldots, y_{g}, z_{1}, \ldots, z_{g}\right| x_{1}^{d_{1}}, \ldots, x_{m}^{d_{m}} \\ \left.x_{1} \cdots x_{m}\left[y_{1}, z_{1}\right] \cdots\left[y_{g}, z_{g}\right]\right\rangle \end{aligned}Γ=⟨x1,…,xm,y1,…,yg,z1,…,zg|x1d1,…,xmdmx1⋯xm[y1,z1]⋯[yg,zg]⟩
define the Euler characteristic
e = 2 2 g i = 1 m ( 1 d i 1 ) e = 2 − 2 g − ∑ i = 1 m   1 − d i − 1 e=2-2g-sum_(i=1)^(m)(1-d_(i)^(-1))e=2-2 g-\sum_{i=1}^{m}\left(1-d_{i}^{-1}\right)e=2−2g−∑i=1m(1−di−1)
Assume e < 0 e < 0 e < 0e<0e<0, so Γ Î“ Gamma\GammaΓ is an oriented, cocompact Fuchsian group. Let G G GGG be a finite group, and let C 1 , , C m C 1 , … , C m C_(1),dots,C_(m)C_{1}, \ldots, C_{m}C1,…,Cm denote conjugacy classes in G G GGG of elements whose orders divide d 1 , , d m d 1 , … , d m d_(1),dots,d_(m)d_{1}, \ldots, d_{m}d1,…,dm,
respectively. The Frobenius formula can be regarded as the g = 0 g = 0 g=0g=0g=0 case of a more general formula for the number of homomorphisms Γ G Γ → G Gamma rarr G\Gamma \rightarrow GΓ→G mapping x i x i x_(i)x_{i}xi to an element of C i C i C_(i)C_{i}Ci for all i i iii,
| Hom { C i } ( Γ , G ) | = | G | 2 g 1 | C 1 | | C m | χ χ ( C 1 ) χ ( C m ) χ ( 1 ) m + 2 g 2 Hom C i ⁡ ( Γ , G ) = | G | 2 g − 1 C 1 ⋯ C m ∑ χ   χ C 1 ⋯ χ C m χ ( 1 ) m + 2 g − 2 |Hom_({C_(i)})(Gamma,G)|=|G|^(2g-1)|C_(1)|cdots|C_(m)|sum_(chi)(chi(C_(1))cdots chi(C_(m)))/(chi(1)^(m+2g-2))\left|\operatorname{Hom}_{\left\{C_{i}\right\}}(\Gamma, G)\right|=|G|^{2 g-1}\left|C_{1}\right| \cdots\left|C_{m}\right| \sum_{\chi} \frac{\chi\left(C_{1}\right) \cdots \chi\left(C_{m}\right)}{\chi(1)^{m+2 g-2}}|Hom{Ci}⁡(Γ,G)|=|G|2g−1|C1|⋯|Cm|∑χχ(C1)⋯χ(Cm)χ(1)m+2g−2
In favorable situations, one can prove that the χ = 1 χ = 1 chi=1\chi=1χ=1 term dominates all the others combined, in which case one has a good estimate for the number of such homomorphisms. Using this, Liebeck and Shalev proved [32, THEOREM 1.5] that if g 2 g ≥ 2 g >= 2g \geq 2g≥2, and G G GGG is a simple of Lie type group of rank r r rrr, then
| Hom ( Γ , G ) | = | G | 1 e + O ( 1 / r ) | Hom ⁡ ( Γ , G ) | = | G | 1 − e + O ( 1 / r ) |Hom(Gamma,G)|=|G|^(1-e+O(1//r))|\operatorname{Hom}(\Gamma, G)|=|G|^{1-e+O(1 / r)}|Hom⁡(Γ,G)|=|G|1−e+O(1/r)
By the same method, employing the character bounds of [28], one obtains the same estimate whenever e e eee is less than some absolute constant, regardless of the value of g g ggg. It would be interesting to know whether this is true in general for e < 0 e < 0 e < 0e<0e<0. Some evidence in favor of this idea is given in [21,33], but for small q q qqq the problem is open.
An interesting geometric consequence of the method of Liebeck-Shalev is that if G _ G _ G_\underline{G}G_ is a simple algebraic group of rank r r rrr and g 2 g ≥ 2 g >= 2g \geq 2g≥2, the morphism G _ 2 g G _ G _ 2 g → G _ G_^(2g)rarrG_\underline{G}^{2 g} \rightarrow \underline{G}G_2g→G_ given by the word [ y 1 , z 1 ] [ y g , z g ] y 1 , z 1 ⋯ y g , z g [y_(1),z_(1)]cdots[y_(g),z_(g)]\left[y_{1}, z_{1}\right] \cdots\left[y_{g}, z_{g}\right][y1,z1]⋯[yg,zg] has all fibers of the same dimension, ( 2 g 1 ) dim G _ ( 2 g − 1 ) dim ⁡ G _ (2g-1)dim G_(2 g-1) \operatorname{dim} \underline{G}(2g−1)dim⁡G_. This has been refined by Avraham Aizenbud and Nir Avni, who proved [1] that for g 373 g ≥ 373 g >= 373g \geq 373g≥373, the fibers of this morphism are reduced and have rational singularities. It would be interesting to extend this to the case of general Fuchsian groups. For instance, does there exist an absolute constant k k k\mathrm{k}k such that for all simple algebraic groups G _ G _ G_\underline{G}G_ and conjugacy classes C _ 1 , , C _ m C _ 1 , … , C _ m C__(1),dots,C__(m)\underline{C}_{1}, \ldots, \underline{C}_{m}C_1,…,C_m with dim C _ 1 + dim ⁡ C _ 1 + dim C__(1)+\operatorname{dim} \underline{C}_{1}+dim⁡C_1+ + dim C _ m > k dim G _ ⋯ + dim ⁡ C _ m > k dim ⁡ G _ cdots+dim C__(m) > k dim G_\cdots+\operatorname{dim} \underline{C}_{m}>k \operatorname{dim} \underline{G}⋯+dim⁡C_m>kdim⁡G_, all fibers of the multiplication morphism C _ 1 × × C _ m G _ C _ 1 × ⋯ × C _ m → G _ C__(1)xx cdots xxC__(m)rarrG_\underline{C}_{1} \times \cdots \times \underline{C}_{m} \rightarrow \underline{G}C_1×⋯×C_m→G_ are reduced with rational singularities The ideas of Glazer-Hendel [11,12] may be applicable.
For g = 1 g = 1 g=1g=1g=1, we can no longer hope for equidimensional fibers, since the generic fiber dimension is dim G _ dim ⁡ G _ dim G_\operatorname{dim} \underline{G}dim⁡G_, while the fiber over the identity element has dimension r + dim G _ r + dim ⁡ G _ r+dim G_r+\operatorname{dim} \underline{G}r+dim⁡G_. However, Zhipeng L u L u Lu\mathrm{Lu}Lu and I proved [20] that for G _ = S L n G _ = S L n G_=SL_(n)\underline{G}=\mathrm{SL}_{n}G_=SLn, all fibers over noncentral elements have dimension G _ G _ G_\underline{G}G_. It would be interesting to know whether this is true for general simple algebraic groups G _ G _ G_\underline{G}G_.

FUNDING

This work was partially supported by the National Science Foundation.

REFERENCES

[1] A. Aizenbud and N. Avni, Representation growth and rational singularities of the moduli space of local systems. Invent. Math. 204 (2016), no. 1, 245-316.
[2] R. Bezrukavnikov, M. Liebeck, A. Shalev, and P. H. Tiep, Character bounds for finite groups of Lie type. Acta Math. 221 (2018), no. 1, 1-57.
[3] P. Biane, Representations of symmetric groups and free probability. Adv. Math. 138 (1998), no. 1, 126-181.
[4] P. Diaconis and M. Shahshahani, Generating a random permutation with random transpositions. Z. Wahrsch. Verw. Gebiete 57 (1981), no. 2, 159-179.
[5] Y. Dvir, Covering properties of permutation groups. In Products of conjugacy classes in groups, pp. 197-221, Lecture Notes in Math. 1112, Springer, Berlin, 1985 .
[6] E. W. Ellers and N. Gordeev, On the conjectures of J. Thompson and O. Ore. Trans. Amer. Math. Soc. 350 (1998), no. 9, 3657-3671.
[7] V. Féray and P. Śniady, Asymptotics of characters of symmetric groups related to Stanley character formula. Ann. of Math. 173 (2011), no. 2, 887-906.
[8] L. Flatto, A. M. Odlyzko, and D. B. Wales, Random shuffles and group representations. Ann. Probab. 13 (1985), no. 1, 154-178.
[9] S. Fomin and N. Lulov, On the number of rim hook tableaux. J. Math. Sci. (N. Y.) 87 (1997), no. 6, 4118-4123.
[10] S. Garion, M. Larsen, and A. Lubotzky, Beauville surfaces and finite simple groups. J. Reine Angew. Math. 666 (2012), 225-243.
[11] I. Glazer and Y. I. Hendel (with an appendix joint with G. Kozma), On singularity properties of convolutions of algebraic morphisms-the general case. 2018, arXiv:1811.09838.
[12] I. Glazer and Y. I. Hendel, On singularity properties of word maps and applications to probabilistic Waring type problems. 2019, arXiv:1912.12556.
[13] D. Gluck, Character value estimates for groups of Lie type. Pacific J. Math. 150 (1991), 279-307.
[14] D. Gluck, Character value estimates for nonsemisimple elements. J. Algebra 155 (1993), no. 1, 221-237.
[15] D. Gluck, Sharper character value estimates for groups of Lie type. J. Algebra 174 (1995), no. 1, 229-266.
[16] R. M. Guralnick, M. Larsen, and P. H. Tiep, Character levels and character bounds. Forum Math. Pi 8 (2020), e2, 81 pp.
[17] R. M. Guralnick, M. Larsen, and P. H. Tiep, Character levels and character bounds for finite classical groups. 2019, arXiv:1904.08070.
[18] C. Hsü, The commutators of the alternating group. Sci. Sinica 14 (1965), 339-342.
[19] D. M. W. H. Husemoller, Ramified coverings of Riemann surfaces. Duke Math. J. 29 (1962), 167-174.
[20] M. Larsen and Z. Lu, Flatness of the commutator map over S L n S L n SL_(n)\mathrm{SL}_{n}SLn. Int. Math. Res. Not. IMRN 2021, no. 8, 5605-5622.
[21] M. Larsen and A. Lubotzky, Representation varieties of Fuchsian groups. In From Fourier analysis and number theory to Radon transforms and geometry, pp. 375-397, Dev. Math. 28, Springer, New York, 2013.
[22] M. Larsen and A. Shalev, Characters of symmetric groups: sharp bounds and applications. Invent. Math. 174 (2008), no. 3, 645-687.
[23] M. Larsen and A. Shalev, Word maps and Waring type problems. J. Amer. Math. Soc. 22 (2009), no. 2, 437-466.
[24] M. Larsen, A. Shalev, and P. H. Tiep, The Waring problem for finite simple groups. Ann. of Math. 174 (2011), no. 3, 1885-1950.
[25] M. Larsen, A. Shalev, and P. H. Tiep, Probabilistic Waring problems for finite simple groups. Ann. of Math. 190 (2019), no. 2, 561-608.
[26] M. Larsen, A. Shalev, and P. H. Tiep, Products of normal subgroups and derangements. 2020, arXiv:2003.12882.
[27] M. Larsen and P. H. Tiep, A refined Waring problem for finite simple groups. Forum Math. Sigma 3 (2015), Paper No. e6, 22 pp.
[28] M. Larsen and P. H. Tiep, Uniform characters bounds for finite classical groups (submitted).
[29] M. W. Liebeck, Character ratios for finite groups of Lie type, and applications. In Finite simple groups: thirty years of the Atlas and beyond, pp. 193-208, Contemp. Math. 694, Amer. Math. Soc., Providence, RI, 2017.
[30] M. W. Liebeck, E. A. O'Brien, A. Shalev, and P. H. Tiep, The Ore conjecture. J. Eur. Math. Soc. 12 (2010), no. 4, 939-1008.
[31] M. W. Liebeck and A. Shalev, Diameters of simple groups: sharp bounds and applications. Ann. of Math. 154 (2001), 383-406.
[32] M. W. Liebeck and A. Shalev, Fuchsian groups, finite simple groups and representation varieties. Invent. Math. 159 (2005), no. 2, 317-367.
[33] M. W. Liebeck, A. Shalev, and P. H. Tiep, Character ratios, representation varieties and random generation of finite groups of Lie type. Adv. Math. 374 (2020), 107386 , 39 p p 107386 , 39 p p 107386,39pp107386,39 \mathrm{pp}107386,39pp.
[34] B. F. Logan and L. A. Shepp, A variational problem for random Young tableaux. Adv. Math. 26 (1977), no. 2, 206-222.
[35] A. Lubotzky, Cayley graphs: eigenvalues, expanders and random walks. In Surveys in combinatorics, 1995 (Stirling), pp. 155-189, London Math. Soc. Lecture Note Ser. 218, Cambridge Univ. Press, Cambridge, 1995.
[36] C. Martinez and E. Zelmanov, Products of powers in finite simple groups. Israel J. Math. 96 (1996), part B, 469-479.
[37] T. W. Müller and J-C. Schlage-Puchta, Character theory of symmetric groups, subgroup growth of Fuchsian groups, and random walks. Adv. Math. 213 (2007), no. 2, 919-982.
[38] A. Rattan and P. Åšniady, Upper bound on the characters of the symmetric groups for balanced Young diagrams and a generalized Frobenius formula. Adv. Math. 218 (2008), 673-695.
[39] Y. Roichman, Upper bound on the characters of the symmetric groups. Invent. Math. 125 (1996), 451-485.
[40] J. Saxl and J. S. Wilson, A note on powers in simple groups. Math. Proc. Cambridge Philos. Soc. 122 (1997), no. 1, 91-94.
[41] A. Shalev, Word maps, conjugacy classes, and a noncommutative Waring-type theorem. Ann. of Math. (2) 170 (2009), no. 3, 1383-1416.
[42] A. Shalev, Conjugacy classes, growth and complexity. In Finite simple groups: thirty years of the atlas and beyond, pp. 209-221, Contemp. Math. 694, Amer. Math. Soc., Providence, RI, 2017.
[43] J. Taylor and P. H. Tiep, Lusztig induction, unipotent supports, and character bounds. Trans. Amer. Math. Soc. 373 (2020), no. 12, 8637-8676.
[44] A. M. Veršik and S. V. Kerov, Asymptotic behavior of the Plancherel measure of the symmetric group and the limit form of Young tableaux (Russian). Dokl. Akad. Nauk SSSR 233 (1977), no. 6, 1024-1027.

MICHAEL J. LARSEN

Department of Mathematics, Indiana University, Bloomington, IN 47405, U.S.A., mjlarsen@indiana.edu

FINITE APPROXIMATIONS AS A TOOL FOR STUDYING TRIANGULATED CATEGORIES

AMNON NEEMAN

Abstract

Small, finite entities are easier and simpler to manipulate than gigantic, infinite ones. Consequently, huge chunks of mathematics are devoted to methods reducing the study of big, cumbersome objects to an analysis of their finite building blocks. The manifestation of this general pattern, in the study of derived and triangulated categories, dates back almost to the beginnings of the subject-more precisely to articles by Illusie in SGA6, way back in the early 1970s.

What is new, at least new in the world of derived and triangulated categories, is that one gets extra mileage from analyzing more carefully and quantifying more precisely just how efficiently one can estimate infinite objects by finite ones. This leads one to the study of metrics on triangulated categories, and of how accurately an object can be approximated by finite objects of bounded size.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 18G80; Secondary 14F08, 55P42

KEYWORDS

Derived categories, triangulated categories, metrics, norms, t-structures

1. INTRODUCTION

In every branch of mathematics, we try to solve complicated problems by reducing to simpler ones, and from antiquity people have used finite approximations to study infinite objects. Naturally, whenever a new field comes into being, one of the first developments is to try to understand what should be the right notion of finiteness in the discipline. Derived and triangulated categories were introduced by Verdier in his PhD thesis in the mid-1960s (although the published version only appeared much later in [38]). Not surprisingly, the idea of studying the finite objects in these categories followed suit soon after, see Illusie [13-15].
Right from the start there was a pervasive discomfort with derived and triangulated categories-the intuition that had been built up, in dealing with concrete categories, mostly fails for triangulated categories. In case the reader is wondering: in the previous sentence the word "concrete" has a precise, technical meaning, and it is an old theorem of Freyd [10,11] that triangulated categories often are not concrete. Further testimony, to the strangeness of derived and triangulated categories, is that it took two decades before the intuitive notion of finiteness, which dates back to Illusie's articles [13-15], was given its correct formal definition. The following may be found in [23, DEFINITION 1.1].
Definition 1.1. Let T T T\mathscr{T}T be a triangulated category with coproducts. An object C T C ∈ T C inTC \in \mathscr{T}C∈T is called compact if Hom ( C , ) Hom ⁡ ( C , − ) Hom(C,-)\operatorname{Hom}(C,-)Hom⁡(C,−) commutes with coproducts. The full subcategory of all compact objects will be denoted by T c T c T^(c)\mathscr{T}^{c}Tc.
Remark 1.2. I have often been asked where the name "compact" came from. In the preprint version of [23], these objects went by a different name, but the (anonymous) referee did not like it. I was given a choice: I was allowed to baptize them either "compact" or "small."
Who was I to argue with a referee?
Once one has a good working definition of what the finite objects ought to be, the next step is to give the right criterion which guarantees that the category has "enough" of them. For triangulated categories, the right definition did not come until [24, DEFINITION 1.7].
Definition 1.3. Let T T T\mathscr{T}T be a triangulated category with coproducts. The category T T T\mathscr{T}T is called compactly generated if every nonzero object X T X ∈ T X inTX \in \mathscr{T}X∈T admits a nonzero map C X C → X C rarr XC \rightarrow XC→X, with C T C ∈ T C inTC \in \mathscr{T}C∈T a compact object.
As the reader may have guessed from the name, compactly generated triangulated categories are those in which it is often possible to reduce general problems to questions about compact objects-which tend to be easier.
All of the above nowadays counts as "classical," meaning that it is two or more decades old and there is already a substantial and diverse literature exploiting the ideas. This article explores the recent developments that arose from trying to understand how efficiently one can approximate arbitrary objects by compact ones. We first survey the results obtained to date. This review is on the skimpy side, partly because there already are other, more expansive published accounts in the literature, but mostly because we want to leave ourselves space to suggest possible directions for future research. Thus the article can be
thought of as having two components: a bare-bone review of what has been achieved to date, occupying Sections 2 to 6 , followed by Section 7 which comprises suggestions of avenues that might merit further development.
Our review presents just enough detail so that the open questions, making up Section 7, can be formulated clearly and comprehensibly, and so that the significance and potential applications of the open questions can be illuminated. This has the unfortunate side effect that we give short shrift to the many deep, substantial contributions, made by numerous mathematicians, which preceded and inspired the work presented here. The author apologizes in advance for this omission, which is the inescapable corollary of page limits. The reader is referred to the other surveys of the subject, where more care is taken to attribute the ideas correctly to their originators, and give credit where credit is due.
We permit ourselves to gloss over difficult technicalities, nonchalantly skating by nuances and subtleties, with only an occasional passing reference to the other surveys or to the research papers for more detail.
The reader wishing to begin with examples and applications, to keep in mind through the forthcoming abstraction, is encouraged to first look at the Introduction to [31].

2. APPROXIMABLE TRIANGULATED CATEGORIES-THE FORMAL DEFINITION AS A VARIANT ON FOURIER SERIES

It is now time to start our review, offering a glimpse of the recent progress that was made by trying to measure how "complicated" an object is, in other words, how far it is from being compact. What follows is sufficiently new for there to be much room for improvement: the future will undoubtedly see cleaner, more elegant, and more general formulations. What is presented here is the current crude state of this emerging field.
Discussion 2.1. This section is devoted to defining approximable triangulated categories, and the definition is technical and at first sight could appear artificial, maybe even forbidding, It might help therefore to motivate it with an analogy.
Let S 1 S 1 S^(1)\mathbb{S}^{1}S1 be the circle, and let M ( S 1 ) M S 1 M(S^(1))M\left(\mathbb{S}^{1}\right)M(S1) be the set of all complex-valued, Lebesguemeasurable functions on S 1 S 1 S^(1)\mathbb{S}^{1}S1. As usual we view S 1 = R / Z S 1 = R / Z S^(1)=R//Z\mathbb{S}^{1}=\mathbb{R} / \mathbb{Z}S1=R/Z as the quotient of its universal cover R R R\mathbb{R}R by the fundamental group Z Z Z\mathbb{Z}Z; this identifies functions on S 1 S 1 S^(1)\mathbb{S}^{1}S1 with periodic functions on R R R\mathbb{R}R with period 1. In particular the function g ( x ) = e 2 π i x g ( x ) = e 2 Ï€ i x g(x)=e^(2pi ix)g(x)=e^{2 \pi i x}g(x)=e2Ï€ix belongs to M ( S 1 ) M S 1 M(S^(1))M\left(\mathbb{S}^{1}\right)M(S1). And, for each Z â„“ ∈ Z â„“inZ\ell \in \mathbb{Z}ℓ∈Z, we have that g ( x ) = e 2 π i x g ( x ) â„“ = e 2 Ï€ i â„“ x g(x)^(â„“)=e^(2pi iâ„“x)g(x)^{\ell}=e^{2 \pi i \ell x}g(x)â„“=e2Ï€iâ„“x also belongs to M ( S 1 ) M S 1 M(S^(1))M\left(\mathbb{S}^{1}\right)M(S1). Given a norm on the space M ( S 1 ) M S 1 M(S^(1))M\left(\mathbb{S}^{1}\right)M(S1), for example, the L p L p L^(p)L^{p}Lp-norm, we can try to approximate arbitrary f M ( S 1 ) f ∈ M S 1 f in M(S^(1))f \in M\left(\mathbb{S}^{1}\right)f∈M(S1) by Laurent polynomials in g g ggg, that is, look for complex numbers { λ C n n } λ â„“ ∈ C ∣ − n ≤ â„“ ≤ n {lambda_(â„“)inC∣-n <= â„“ <= n}\left\{\lambda_{\ell} \in \mathbb{C} \mid-n \leq \ell \leq n\right\}{λℓ∈C∣−n≤ℓ≤n} such that
f ( x ) = n n λ g ( x ) p = f ( x ) = n n λ e 2 π i x p < ε f ( x ) − ∑ â„“ = − n n   λ â„“ g ( x ) â„“ p = f ( x ) − ∑ â„“ = − n n   λ â„“ e 2 Ï€ i â„“ x p < ε ||f(x)-sum_(â„“=-n)^(n)lambda_(â„“)g(x)^(â„“)||_(p)=||f(x)-sum_(â„“=-n)^(n)lambda_(â„“)e^(2pi iâ„“x)||_(p) < epsi\left\|f(x)-\sum_{\ell=-n}^{n} \lambda_{\ell} g(x)^{\ell}\right\|_{p}=\left\|f(x)-\sum_{\ell=-n}^{n} \lambda_{\ell} e^{2 \pi i \ell x}\right\|_{p}<\varepsilon∥f(x)−∑ℓ=−nnλℓg(x)ℓ∥p=∥f(x)−∑ℓ=−nnλℓe2Ï€iâ„“x∥p<ε
with ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0 small. This leads us to the familiar territory of Fourier series.
Now imagine trying to do the same, but replacing M ( S 1 ) M S 1 M(S^(1))M\left(\mathbb{S}^{1}\right)M(S1) by a triangulated category. Given a triangulated category T T T\mathscr{T}T, which we assume to have coproducts, we would like to pretend to do Fourier analysis on it. We would need to choose:
(1) Some analog of the function g ( x ) = e 2 π i x g ( x ) = e 2 Ï€ i x g(x)=e^(2pi ix)g(x)=e^{2 \pi i x}g(x)=e2Ï€ix. Our replacement for this will be to choose a compact generator G T G ∈ T G inTG \in \mathscr{T}G∈T. Recall that a compact generator is a compact object G T G ∈ T G inTG \in \mathscr{T}G∈T such that every nonzero object X T X ∈ T X inTX \in \mathscr{T}X∈T admits a nonzero map G [ i ] X G [ i ] → X G[i]rarr XG[i] \rightarrow XG[i]→X for some i Z i ∈ Z i inZi \in \mathbb{Z}i∈Z.
(2) We need to choose something like a metric, the analog of the L p L p L^(p)L^{p}Lp-norm on M ( S 1 ) M S 1 M(S^(1))M\left(\mathbb{S}^{1}\right)M(S1). For us this will be done by picking a t-structure ( T 0 , T 0 ) T ≤ 0 , T ≥ 0 (T <= 0,T^( >= 0))\left(\mathscr{T} \leq 0, \mathscr{T}^{\geq 0}\right)(T≤0,T≥0) on T T T\mathscr{T}T. The heuristic is that we will view a morphism E F E → F E rarr FE \rightarrow FE→F in T T T\mathscr{T}T as "short" if, in the triangle E F D E → F → D E rarr F rarr DE \rightarrow F \rightarrow DE→F→D, the object D D DDD belongs to T n T ≤ − n T^( <= -n)\mathscr{T}^{\leq-n}T≤−n for large n n nnn. We will come back to this in Discussion 6.10.
(3) We need to have an analog of the construction that passes, from the function g ( x ) = e 2 π i x g ( x ) = e 2 Ï€ i x g(x)=e^(2pi ix)g(x)=e^{2 \pi i x}g(x)=e2Ï€ix and the integer n > 0 n > 0 n > 0n>0n>0, to the vector space of trigonometric Laurent polynomials = n n λ e 2 π i x ∑ â„“ = − n n   λ â„“ e 2 Ï€ i â„“ x sum_(â„“=-n)^(n)lambda_(â„“)e^(2pi iâ„“x)\sum_{\ell=-n}^{n} \lambda_{\ell} e^{2 \pi i \ell x}∑ℓ=−nnλℓe2Ï€iâ„“x.
As it happens our solution to (3) is technical. We need a recipe that begins with the object G G GGG and the integer n > 0 n > 0 n > 0n>0n>0, and proceeds to cook up a collection of more objects. We ask the reader to accept it as a black box, with only a sketchy explanation just before Remark 2.3.
Black Box 2.2. Let T T T\mathscr{T}T be a triangulated category and let G T G ∈ T G inTG \in \mathscr{T}G∈T be an object. Let n > 0 n > 0 n > 0n>0n>0 be an integer. We will have occasion to refer to the following four full subcategories of T T T\mathscr{T}T :
(1) The subcategory G n T ⟨ G ⟩ n ⊂ T (:G:)_(n)subT\langle G\rangle_{n} \subset \mathscr{T}⟨G⟩n⊂T is defined unconditionally, and if T T T\mathscr{T}T has coproducts one can also define the larger subcategory G ¯ n ⟨ G ⟩ ¯ n bar((:G:))_(n)\overline{\langle G\rangle}_{n}⟨G⟩¯n. Both of these subcategories are classical, the reader can find the subcategory G n ⟨ G ⟩ n (:G:)_(n)\langle G\rangle_{n}⟨G⟩n in Bondal and Van den Bergh [6, THE DISCUSSION BETWEEN LEMMA 2.2.2 AND DEFINITION 2.2.3], and the subcategory G ¯ ⟨ G ⟩ ¯ bar((:G:))\overline{\langle G\rangle}⟨G⟩¯ in [6, THE DISCUSSION BETWEEN DEFINITION 2.2.3 AND PROPOSITION 2.2.4].
(2) If the category T T T\mathscr{T}T has coproducts, we will also have occasion to consider the full subcategory G ¯ ( , n ] ⟨ G ⟩ ¯ ( − ∞ , n ] bar((:G:))(-oo,n]\overline{\langle G\rangle}(-\infty, n]⟨G⟩¯(−∞,n]. Once again this category is classical (although the name is not). The reader can find it in Alonso, Jeremías, and Souto [1], where it would go by the name "the cocomplete pre-aisle generated by G [ n ] G [ − n ] G[-n]G[-n]G[−n] ".
(3) Once again assume that T T T\mathscr{T}T has coproducts. Then we will also look at the full
Below we give a vague description of what is going on in these constructions; but when it comes to the technicalities, we ask the reader to either accept these as black boxes, or refer to [29, REMINDER 0.8 (VII), (XI) AND (XII)] for detail. We mention that there is a slight clash of notation in the literature: what we call G ¯ ⟨ G ⟩ ¯ bar((:G:))\overline{\langle G\rangle}⟨G⟩¯ in (1), following Bondal and Van den Bergh,
goes by a different name in [29, REMINDER 0.8 (XI)]. The name it goes by there is the case A = A = − ∞ A=-ooA=-\inftyA=−∞ and B = B = ∞ B=ooB=\inftyB=∞ of the more general subcategory G ¯ n [ A , B ] ⟨ G ⟩ ¯ n [ A , B ] bar((:G:))_(n)^([A,B])\overline{\langle G\rangle}_{n}^{[A, B]}⟨G⟩¯n[A,B].
Now for the vague explanation of what goes on in (1), (2), and (3) above: in a triangulated category T T T\mathscr{T}T, there are not many ways to build new objects out of old ones. One can shift objects, form direct summands, form finite direct sums (or infinite ones if coproducts exist), and one can form extensions. In the categories G n ⟨ G ⟩ n (:G:)_(n)\langle G\rangle_{n}⟨G⟩n and G n ¯ ⟨ G ⟩ n ¯ bar((:G:)_(n))\overline{\langle G\rangle_{n}}⟨G⟩n¯ of (1), there is a bound on the number of allowed extensions, and the difference between the two is whether infinite coproducts are allowed. In the category G ¯ ( , n ] ⟨ G ⟩ ¯ ( − ∞ , n ] bar((:G:))^((-oo,n])\overline{\langle G\rangle}{ }^{(-\infty, n]}⟨G⟩¯(−∞,n] of (2), the bound is on the permitted shifts. And in the category G ¯ n [ n , n ] ⟨ G ⟩ ¯ n [ − n , n ] bar((:G:))_(n)^([-n,n])\overline{\langle G\rangle}_{n}^{[-n, n]}⟨G⟩¯n[−n,n] of (3), both the shifts allowed and the number of extensions permitted are restricted.
Remark 2.3. The reader should note that an example would not be illuminating, the categories G n , G ¯ n , G ¯ ( , n ] ⟨ G ⟩ n , ⟨ G ⟩ ¯ n , ⟨ G ⟩ ¯ ( − ∞ , n ] (:G:)_(n), bar((:G:))_(n), bar((:G:))^((-oo,n])\langle G\rangle_{n}, \overline{\langle G\rangle}_{n}, \overline{\langle G\rangle}^{(-\infty, n]}⟨G⟩n,⟨G⟩¯n,⟨G⟩¯(−∞,n], and G n [ n , n ] ¯ ⟨ G ⟩ n [ − n , n ] ¯ bar((:G:)_(n)^([-n,n]))\overline{\langle G\rangle_{n}^{[-n, n]}}⟨G⟩n[−n,n]¯ are not usually overly computable. For example, let R R RRR be an associative ring, and let T = D ( R ) T = D ( R ) T=D(R)\mathscr{T}=\mathbf{D}(R)T=D(R) be the unbounded derived category of complexes of left R R RRR-modules. The object R T R ∈ T R inTR \in \mathscr{T}R∈T, that is, the complex which is R R RRR in degree zero and vanishes in all other degrees, is a compact generator for T = D ( R ) T = D ( R ) T=D(R)\mathscr{T}=\mathbf{D}(R)T=D(R).
But if we wonder what the categories R n , R ¯ n , R ¯ ( , n ] ⟨ R ⟩ n , ⟨ R ⟩ ¯ n , ⟨ R ⟩ ¯ ( − ∞ , n ] (:R:)_(n), bar((:R:))_(n), bar((:R:))^((-oo,n])\langle R\rangle_{n}, \overline{\langle R\rangle}_{n}, \overline{\langle R\rangle}^{(-\infty, n]}⟨R⟩n,⟨R⟩¯n,⟨R⟩¯(−∞,n], and R ¯ n [ n , n ] ⟨ R ⟩ ¯ n [ − n , n ] bar((:R:))_(n)^([-n,n])\overline{\langle R\rangle}_{n}^{[-n, n]}⟨R⟩¯n[−n,n] might turn out to be, only the category R ¯ ( , n ] ⟨ R ⟩ ¯ ( − ∞ , n ] bar((:R:))^((-oo,n])\overline{\langle R\rangle}^{(-\infty, n]}⟨R⟩¯(−∞,n] is straightforward: it is the category of all cochain complexes whose cohomology vanishes in degrees > n > n > n>n>n. The three categories R n , R n ¯ ⟨ R ⟩ n , ⟨ R ⟩ n ¯ (:R:)_(n), bar((:R:)_(n))\langle R\rangle_{n}, \overline{\langle R\rangle_{n}}⟨R⟩n,⟨R⟩n¯, and R ¯ n [ n , n ] ⟨ R ⟩ ¯ n [ − n , n ] bar((:R:))_(n)^([-n,n])\overline{\langle R\rangle}_{n}^{[-n, n]}⟨R⟩¯n[−n,n] are mysterious in general. In fact, the computation of G n ⟨ G ⟩ n (:G:)_(n)\langle G\rangle_{n}⟨G⟩n is the subject of conjectures that have attracted much interest. We will say a tiny bit about theorems in this direction in Section 4, and will mention one of the active, open conjectures in the discussion between Definition 7.7 and Problem 7.8.
Remark 2.4. In the definition of approximable triangulated categories, which is about to come, the category G n ¯ [ n , n ] ⟨ G ⟩ n ¯ [ − n , n ] bar((:G:)_(n))[-n,n]\overline{\langle G\rangle_{n}}[-n, n]⟨G⟩n¯[−n,n] will play the role of the replacement for the vector space of trigonometric Laurent polynomials of degree n ≤ n <= n\leq n≤n, which came up in the desiderata of Discussion 2.1(3). The older categories G n , G ¯ n ⟨ G ⟩ n , ⟨ G ⟩ ¯ n (:G:)_(n), bar((:G:))_(n)\langle G\rangle_{n}, \overline{\langle G\rangle}_{n}⟨G⟩n,⟨G⟩¯n, and G ¯ ( , n ] ⟨ G ⟩ ¯ ( − ∞ , n ] bar((:G:))(-oo,n]\overline{\langle G\rangle}(-\infty, n]⟨G⟩¯(−∞,n] will be needed later in the article.
Remark 2.5. Let us return to the heuristics of Discussion 2.1. Assume we have chosen the t-structure ( T 0 , T 0 ) T ≤ 0 , T 0 (T <= 0,T^(0))\left(\mathscr{T} \leq 0, \mathscr{T}^{0}\right)(T≤0,T0) as in Discussion 2.1(2), which we think of as our replacement for the L p L p L^(p)L^{p}Lp-norm on M ( S 1 ) M S 1 M(S^(1))M\left(\mathbb{S}^{1}\right)M(S1). And we have also chosen a compact generator G T G ∈ T G inTG \in \mathscr{T}G∈T as in Discussion 2.1(1), which we think of as the analog of the exponential function g ( x ) = e 2 π i x g ( x ) = e 2 Ï€ i x g(x)=e^(2pi ix)g(x)=e^{2 \pi i x}g(x)=e2Ï€ix. We have declared that the subcategories G n [ n , n ] ¯ ⟨ G ⟩ n [ − n , n ] ¯ bar((:G:)_(n)^([-n,n]))\overline{\langle G\rangle_{n}^{[-n, n]}}⟨G⟩n[−n,n]¯ will be our replacement for the vector space of trigonometric Laurent polynomials of degree n ≤ n <= n\leq n≤n, as in Discussion 2.1(3). It is now time to start approximating functions by trigonometric Laurent polynomials.
Let us therefore assume we start with some object F T F ∈ T F inTF \in \mathscr{T}F∈T, and find a good approximation of it by the object E G ¯ m [ m , m ] E ∈ ⟨ G ⟩ ¯ m [ − m , m ] E in bar((:G:))_(m)^([-m,m])E \in \overline{\langle G\rangle}_{m}^{[-m, m]}E∈⟨G⟩¯m[−m,m], meaning that we find a morphism E F E → F E rarr FE \rightarrow FE→F such that, in the triangle E F D E → F → D E rarr F rarr DE \rightarrow F \rightarrow DE→F→D, the object D D DDD belongs to T M T ≤ − M T <= -M\mathscr{T} \leq-MT≤−M for some suitably large M M MMM.
Now we can try to iterate, and find a good approximation for D D DDD. Thus we can look for a morphism E D E ′ ′ → D E^('')rarr DE^{\prime \prime} \rightarrow DE′′→D, with E G n ¯ n , n ] E ′ ′ ∈ ⟨ G ⟩ n ¯ n , n ] E^('')in bar((:G:)_(n))_(n,n])E^{\prime \prime} \in \overline{\langle G\rangle_{n}} \underset{n, n]}{ }E′′∈⟨G⟩n¯n,n], and such that in the triangle E D D E ′ ′ → D → D ′ E^('')rarr D rarrD^(')E^{\prime \prime} \rightarrow D \rightarrow D^{\prime}E′′→D→D′ the
object D D ′ D^(')D^{\prime}D′ belongs to T N T ≤ − N T <= -N\mathscr{T} \leq-NT≤−N, with N > M N > M N > MN>MN>M even more enormous than M M MMM. Can we combine these to improve our initial approximation of F F FFF ?
To do this, let us build up the octahedron on the composable morphisms F F → F rarrF \rightarrowF→ D D D → D ′ D rarrD^(')D \rightarrow D^{\prime}D→D′. We end up with a diagram where the rows and columns are triangles
and in particular the triangle E F D E ′ → F → D ′ E^(')rarr F rarrD^(')E^{\prime} \rightarrow F \rightarrow D^{\prime}E′→F→D′ gives that E E ′ E^(')E^{\prime}E′ is an even better approximation of F F FFF than E E EEE was. We are therefore interested in knowing if the triangle E E E E → E ′ → E ′ ′ E rarrE^(')rarrE^('')E \rightarrow E^{\prime} \rightarrow E^{\prime \prime}E→E′→E′′, coupled with the fact that E G m ¯ m , m ] E ∈ ⟨ G ⟩ m ¯ m , m ] E in bar((:G:)_(m))_(m,m])E \in \overline{\langle G\rangle_{m}} \underset{m, m]}{ }E∈⟨G⟩m¯m,m] and E G n ¯ [ n , n ] E ′ ′ ∈ ⟨ G ⟩ n ¯ [ − n , n ] E^('')in bar((:G:)_(n))^([-n,n])E^{\prime \prime} \in \overline{\langle G\rangle_{n}}{ }^{[-n, n]}E′′∈⟨G⟩n¯[−n,n], gives any information about where E E ′ E^(')E^{\prime}E′ might lie with respect to the construction of Black Box 2.2(3). Hence it is useful to know the following.
Facts 2.6. Let T T T\mathscr{T}T be a triangulated category with coproducts. The construction of Black Box 2.2(3) satisfies

G ¯ n + 1 [ n 1 , n + 1 ] ⟨ G ⟩ ¯ n + 1 [ − n − 1 , n + 1 ] bar((:G:))_(n+1)^([-n-1,n+1])\overline{\langle G\rangle}_{n+1}^{[-n-1, n+1]}⟨G⟩¯n+1[−n−1,n+1].
(2) Given an exact triangle E E E E → E ′ → E ′ ′ E rarrE^(')rarrE^('')E \rightarrow E^{\prime} \rightarrow E^{\prime \prime}E→E′→E′′, with E G ¯ m [ m , m ] E ∈ ⟨ G ⟩ ¯ m [ − m , m ] E in bar((:G:))_(m)^([-m,m])E \in \overline{\langle G\rangle}_{m}^{[-m, m]}E∈⟨G⟩¯m[−m,m] and E G ¯ n [ n , n ] E ′ ′ ∈ ⟨ G ⟩ ¯ n [ − n , n ] E^('')in bar((:G:))_(n)^([-n,n])E^{\prime \prime} \in \overline{\langle G\rangle}_{n}^{[-n, n]}E′′∈⟨G⟩¯n[−n,n], it follows that E G ¯ m + n [ m n , m + n ] E ′ ∈ ⟨ G ⟩ ¯ m + n [ − m − n , m + n ] E^(')in bar((:G:))_(m+n)^([-m-n,m+n])E^{\prime} \in \overline{\langle G\rangle}_{m+n}^{[-m-n, m+n]}E′∈⟨G⟩¯m+n[−m−n,m+n].
Combining Remark 2.5 with Facts 2.6 allows us to improve approximations through iteration. Hence part (2) of the definition below becomes natural, it iterates to provide arbitrarily good approximations.
Definition 2.7. Let T T T\mathscr{T}T be a triangulated category with coproducts. It is approximable if there exist a t-structure ( T 0 , T 0 ) ( T ≤ 0 , T ≥ 0 ) (T <= 0,T >= 0)(\mathscr{T} \leq 0, \mathscr{T} \geq 0)(T≤0,T≥0), a compact generator G T G ∈ T G inTG \in \mathscr{T}G∈T, and an integer n > 0 n > 0 n > 0n>0n>0 such that
(1) G G GGG belongs to T n T ≤ n T <= n\mathscr{T} \leq nT≤n and Hom ( G , T n ) = 0 Hom ⁡ ( G , T ≤ − n ) = 0 Hom(G,T <= -n)=0\operatorname{Hom}(G, \mathscr{T} \leq-n)=0Hom⁡(G,T≤−n)=0;
(2) Every object X T 0 X ∈ T ≤ 0 X inT <= 0X \in \mathscr{T} \leq 0X∈T≤0 admits an exact triangle E X D E → X → D E rarr X rarr DE \rightarrow X \rightarrow DE→X→D with E E ∈ E inE \inE∈ G n [ n , n ] ¯ ⟨ G ⟩ n [ − n , n ] ¯ bar((:G:)_(n)[-n,n])\overline{\langle G\rangle_{n}[-n, n]}⟨G⟩n[−n,n]¯ and with D T 1 D ∈ T ≤ − 1 D inT <= -1D \in \mathscr{T} \leq-1D∈T≤−1.
Remark 2.8. While part (2) of Definition 2.7 comes motivated by the analogy with Fourier analysis, part (1) of the definition seems random. It requires the t t ttt-structure, which is our replacement for the L p L p L^(p)L^{p}Lp-norm, to be compatible with the compact generator, which is the analog of g ( x ) = e 2 π i x g ( x ) = e 2 Ï€ i x g(x)=e^(2pi ix)g(x)=e^{2 \pi i x}g(x)=e2Ï€ix. As the reader will see in Proposition 5.5, this has the effect of uniquely specifying the t t ttt-structure (up to equivalence). So maybe a better parallel would be to fix our norm to be a particularly nice one, for example, the L 2 L 2 L^(2)L^{2}L2-norm on M ( S 1 ) M S 1 M(S^(1))M\left(\mathbb{S}^{1}\right)M(S1).
Let me repeat myself: as with all new mathematics, Definition 2.7 should be viewed as provisional. In the remainder of this survey, we will discuss the applications as they now
stand, to highlight the power of the methods. But I would not be surprised in the slightest if future applications turn out to require modifications, and/or generalizations, of the definitions and of the theorems that have worked so far.

3. EXAMPLES OF APPROXIMABLE TRIANGULATED CATEGORIES

In Section 1 we gave the definition of approximable triangulated categories. The definition combines old, classical ingredients (t-structures and compact generators) with a new

theory is nonempty: we need to produce examples, categories people care about which satisfy the definition of approximability. The current section is devoted to the known examples of approximable triangulated categories. We repeat what we have said before: the subject is in its infancy, there could well be many more examples out there.
Example 3.1. Let T T T\mathscr{T}T be a triangulated category with coproducts. If G T G ∈ T G inTG \in \mathscr{T}G∈T is a compact generator such that Hom ( G , G [ i ] ) = 0 Hom ⁡ ( G , G [ i ] ) = 0 Hom(G,G[i])=0\operatorname{Hom}(G, G[i])=0Hom⁡(G,G[i])=0 for all i > 0 i > 0 i > 0i>0i>0, then the category T T T\mathscr{T}T is approximable.
This example turns out to be easy, the reader is referred to [29, EXAMPLE 3.3] for the (short) proof. Special cases include
(1) T = D ( R M o d ) T = D ( R − M o d ) T=D(R-Mod)\mathscr{T}=\mathbf{D}(R-\mathrm{Mod})T=D(R−Mod), where R R RRR is a dga with H i ( R ) = 0 H i ( R ) = 0 H^(i)(R)=0H^{i}(R)=0Hi(R)=0 for i > 0 i > 0 i > 0i>0i>0;
(2) The homotopy category of spectra.
Example 3.2. If X X XXX is a quasicompact, separated scheme, then the category D q c ( X ) D q c ( X ) D_(qc)(X)\mathbf{D}_{\mathbf{q c}}(X)Dqc(X) is approximable. We remind the reader of the traditional notation being used here: the category D ( X ) D ( X ) D(X)\mathbf{D}(X)D(X) is the unbounded derived category of complexes of sheaves of O X O X O_(X)\mathscr{O}_{X}OX-modules, and the full subcategory D q c ( X ) D ( X ) D q c ( X ) ⊂ D ( X ) D_(qc)(X)subD(X)\mathbf{D}_{\mathbf{q c}}(X) \subset \mathbf{D}(X)Dqc(X)⊂D(X) has for objects the complexes with quasicoherent cohomology.
The proof of the approximability of D q c ( X ) D q c ( X ) D_(qc)(X)\mathbf{D}_{\mathbf{q c}}(X)Dqc(X) is not trivial. The category has a standard t t ttt-structure, that part is easy. The existence of a compact generator G G GGG needs proof, it may be found in Bondal and Van den Bergh [6, THEOREM 3.1.1(II)]. Their proof is not constructive, it is only an existence proof, but it does give enough information to deduce that part (1) of Definition 2.7 is satisfied by every compact generator (indeed, it is satisfied by every compact object). See [6, THEOREM 3.1.1(I)]. But it is a challenge to show that we may choose a compact generator G G GGG and an integer n > 0 n > 0 n > 0n>0n>0 in such a way that Definition 2.7(2) is satisfied.
If we further assume that X X XXX is of finite type over a noetherian ring R R RRR, then the (relatively intricate) proof of the approximability of D q c ( X ) D q c ( X ) D_(qc)(X)\mathbf{D}_{\mathbf{q c}}(X)Dqc(X) occupies [33, SECTIONS 4 AND 5]. The little trick, that extends the result to all quasicompact and separated X X XXX, was not observed until later: it appears in [29, LEMMA 3.5].
Example 3.3. It is a theorem that, under mild hypotheses, the recollement of any two approximable triangulated categories is approximable. To state the "mild hypotheses" precisely: suppose we are given a recollement of triangulated categories

with R R R\mathscr{R}R and T T T\mathscr{T}T approximable. Assume further that the category S S S\mathscr{S}S is compactly generated, and any compact object H S H ∈ S H inSH \in \mathscr{S}H∈S has the property that Hom ( H , H [ i ] ) = 0 Hom ⁡ ( H , H [ i ] ) = 0 Hom(H,H[i])=0\operatorname{Hom}(H, H[i])=0Hom⁡(H,H[i])=0 for i 0 i ≫ 0 i≫0i \gg 0i≫0. Then the category S S S\mathscr{S}S is also approximable.
The reader can find the proof in [7, THEOREM 4.1], it is the main result in the paper. The bulk of the article is devoted to developing the machinery necessary to prove the theoremhence it is worth noting that this machinery has since demonstrated usefulness in other contexts, see the subsequent articles [27,28].
There is a beautiful theory of noncommutative schemes, and a rich literature studying them. And many of the interesting examples of such schemes are obtained as recollements of ordinary schemes, or of admissible pieces of them. Thus the theorem that recollements of approximable triangulated categories are approximable gives a wealth of new examples of approximable triangulated categories.
Since this ICM is being held in St. Petersburg, it would be remiss not to mention that the theory of noncommutative algebraic geometry, in the sense of the previous paragraph, is a subject to which Russian mathematicians have contributed a vast amount. The seminal work of Bondal, Kontsevich, Kuznetsov, Lunts, and Orlov immediately springs to mind. For a beautiful introduction to the field, the reader might wish to look at the early sections of Orlov [34]. The later sections prove an amazing new theorem, but the early ones give a lovely survey of the background. In fact, the theory sketched in this survey was born when I was trying to read and understand Orlov's beautiful article.

4. APPLICATIONS: STRONG GENERATION

We begin by reminding the reader of a classical definition, going back to Bondal and Van den Bergh [6].
Definition 4.1. Let T T T\mathscr{T}T be triangulated category. An object G T G ∈ T G inTG \in \mathscr{T}G∈T is called a strong generator if there exists an integer > 0 ℓ > 0 ℓ > 0\ell>0ℓ>0 with T = G T = ⟨ G ⟩ ℓ T=(:G:)_(ℓ)\mathscr{T}=\langle G\rangle_{\ell}T=⟨G⟩ℓ, where the notation is as in Black Box 2.2(1). The category T T T\mathscr{T}T is called regular or strongly generated if it contains a strong generator.
The first application of approximability is the proof of the following two theorems.
Theorem 4.2. Let X X XXX be a quasicompact, separated scheme. The derived category of perfect complexes on X X XXX, denoted here by D perf ( X ) D perf  ( X ) D^("perf ")(X)\mathbf{D}^{\text {perf }}(X)Dperf (X), is regular if and only if X X XXX has a cover by open subsets Spec ( R i ) X Spec ⁡ R i ⊂ X Spec(R_(i))sub X\operatorname{Spec}\left(R_{i}\right) \subset XSpec⁡(Ri)⊂X, with each R i R i R_(i)R_{i}Ri of finite global dimension.
Remark 4.3. If X X XXX is noetherian and separated, then Theorem 4.2 specializes to saying that D perf ( X ) D perf  ( X ) D^("perf ")(X)\mathbf{D}^{\text {perf }}(X)Dperf (X) is regular if and only if X X XXX is regular and finite-dimensional. Hence the terminology.
Theorem 4.4. Let X X XXX be a noetherian, separated, finite-dimensional, quasiexcellent scheme. Then the category D b ( Coh ( X ) ) D b ( Coh ⁡ ( X ) ) D^(b)(Coh(X))\mathbf{D}^{b}(\operatorname{Coh}(X))Db(Coh⁡(X)), the bounded derived category of coherent sheaves on X X XXX, is always regular.
Remark 4.5. The reader is referred to [33] and to Aoki [4] for the proofs of Theorems 4.2 and 4.4. More precisely, for Theorem 4.2 see [33, THEOREM 0.5]. About Theorem 4.4: if we add the assumption that every closed subvariety of X X XXX admits a regular alteration then the result may be found in [33, THEOREM 0.15], but Aoki [4] found a lovely argument that allowed him to extend the statement to all quasiexcellent X X XXX.
There is a rich literature on strong generation, with beautiful papers by many authors. In the introduction to [33], as well as in [26] and [31, sEction 7], the reader can find an extensive discussion of (some of) this fascinating work and of the way Theorems 4.2 and 4.4 compare to the older literature. For a survey taking an entirely different tack, see Minami [22], which places in historical perspective a couple of the key steps in the proofs that [33] gives for Theorems 4.2 and 4.4.
Since all of this is now well documented in the published literature, let us focus the remainder of the current survey on the other applications of approximability. Those are all still in preprint form, see [27-29], although there are (published) surveys in [31, SECTIONs 8 AND 9] and in [30]. Those surveys are fuller and more complete than the sketchy one we are about to embark on. As we present the material, we will feel free to refer the reader to the more extensive surveys whenever we deem it appropriate.

5. THE FREEDOM IN THE CHOICE OF COMPACT GENERATOR AND T-STRUCTURE

Definition 2.7 tells us that a triangulated category T T T\mathscr{T}T with coproducts is approximable if there exist, in T T T\mathscr{T}T, a compact generator G G GGG and a t-structure ( T 0 , T 0 ) ( T ≤ 0 , T ≥ 0 ) (T <= 0,T >= 0)(\mathscr{T} \leq 0, \mathscr{T} \geq 0)(T≤0,T≥0) satisfying some properties. The time has come to explore just how free we are in the choice of the compact generator and of the t t ttt-structure. To address this question we begin by formulating:
Definition 5.1. Let T T T\mathscr{T}T be a triangulated category. Then two t-structures ( T 1 0 , T 1 0 ) T 1 ≤ 0 , T 1 ≥ 0 (T_(1)^( <= 0),T_(1)^( >= 0))\left(\mathscr{T}_{1}^{\leq 0}, \mathscr{T}_{1}^{\geq 0}\right)(T1≤0,T1≥0) and ( T 2 0 , T 2 0 ) T 2 ≤ 0 , T 2 ≥ 0 (T_(2)^( <= 0),T_(2)^( >= 0))\left(\mathscr{T}_{2}^{\leq 0}, \mathscr{T}_{2}^{\geq 0}\right)(T2≤0,T2≥0) are declared equivalent if there exists an integer n > 0 n > 0 n > 0n>0n>0 such that
T 1 n T 2 0 T 1 n T 1 ≤ − n ⊂ T 2 ≤ 0 ⊂ T 1 ≤ n T_(1)^( <= -n)subT_(2)^( <= 0)subT_(1)^( <= n)\mathscr{T}_{1}^{\leq-n} \subset \mathscr{T}_{2}^{\leq 0} \subset \mathscr{T}_{1}^{\leq n}T1≤−n⊂T2≤0⊂T1≤n
Discussion 5.2. Let T T T\mathscr{T}T be a triangulated category with coproducts. If G T G ∈ T G inTG \in \mathscr{T}G∈T is a compact object and G ¯ ( , 0 ] ⟨ G ⟩ ¯ ( − ∞ , 0 ] bar((:G:))^((-oo,0])\overline{\langle G\rangle}{ }^{(-\infty, 0]}⟨G⟩¯(−∞,0] is as in Black Box 2.2(2), then Alonso, Jeremías, and Souto [1, Ñ‚HEoREM A.1], building on the work of Keller and Vossieck [16], teaches us that there is a unique t-structure ( T 0 , T 0 ) T ≤ 0 , T ≥ 0 (T^( <= 0),T^( >= 0))\left(\mathscr{T}^{\leq 0}, \mathscr{T}^{\geq 0}\right)(T≤0,T≥0) with T 0 = G ¯ ( , n ] T ≤ 0 = ⟨ G ⟩ ¯ ( − ∞ , n ] T <= 0= bar((:G:))(-oo,n]\mathscr{T} \leq 0=\overline{\langle G\rangle}(-\infty, n]T≤0=⟨G⟩¯(−∞,n]. We will call this the t t ttt-structure generated by G G GGG, and denote it ( T G 0 , T G 0 ) T G ≤ 0 , T G ≥ 0 (T_(G)^( <= 0),T_(G)^( >= 0))\left(\mathscr{T}_{G}^{\leq 0}, \mathscr{T}_{G}^{\geq 0}\right)(TG≤0,TG≥0).
In Black Box 2.2(2) we asked the reader to accept, as a black box, the construction passing from an object G T G ∈ T G inTG \in \mathscr{T}G∈T to the subcategory G ¯ ( , 0 ] ⟨ G ⟩ ¯ ( − ∞ , 0 ] bar((:G:))^((-oo,0])\overline{\langle G\rangle}{ }^{(-\infty, 0]}⟨G⟩¯(−∞,0]. If G G GGG is compact, then [ 1 [ 1 [1[1[1, THEoREM A.1] allows us to express this as T G 0 T G ≤ 0 T_(G)^( <= 0)\mathscr{T}_{G}^{\leq 0}TG≤0 for a unique t t ttt-structure. We ask the reader to accept on faith that:
Lemma 5.3. If G G GGG and H H HHH are two compact generators for the triangulated category T T T\mathscr{T}T, then the two t t ttt-structures ( T G 0 , T G 0 ) T G ≤ 0 , T G ≥ 0 (T_(G)^( <= 0),T_(G)^( >= 0))\left(\mathscr{T}_{G}^{\leq 0}, \mathscr{T}_{G}^{\geq 0}\right)(TG≤0,TG≥0) and ( T H 0 , T H 0 ) T H ≤ 0 , T H ≥ 0 (T_(H)^( <= 0),T_(H)^( >= 0))\left(\mathscr{T}_{H}^{\leq 0}, \mathscr{T}_{H}^{\geq 0}\right)(TH≤0,TH≥0) are equivalent as in Definition 5.2.
As it happens, the proof of Lemma 5.3 is easy, the interested reader can find it in [29, Remark 0.15]. And Lemma 5.3 leads us to:
Definition 5.4. Let T T T\mathscr{T}T be a triangulated category in which there exists a compact generator. We define the preferred equivalence class of t t ttt-structures as follows: a t t ttt-structure belongs to the preferred equivalence class if it is equivalent to ( T G 0 , T G 0 ) T G ≤ 0 , T G ≥ 0 (T_(G)^( <= 0),T_(G)^( >= 0))\left(\mathscr{T}_{G}^{\leq 0}, \mathscr{T}_{G}^{\geq 0}\right)(TG≤0,TG≥0) for some compact generator G T G ∈ T G inTG \in \mathscr{T}G∈T, and by Lemma 5.3 it is equivalent to ( T H 0 , T H 0 ) T H ≤ 0 , T H ≥ 0 (T_(H)^( <= 0),T_(H)^( >= 0))\left(\mathscr{T}_{H}^{\leq 0}, \mathscr{T}_{H}^{\geq 0}\right)(TH≤0,TH≥0) for every compact generator H H HHH.
The following is also not too hard, and may be found in [29, PROPOSITIONS 2.4 AND 2.6].
Proposition 5.5. Let T T T\mathscr{T}T be an approximable triangulated category. Then for any t t ttt-structure ( T 0 , T 0 ) T ≤ 0 , T ≥ 0 (T <= 0,T^( >= 0))\left(\mathscr{T} \leq 0, \mathscr{T}^{\geq 0}\right)(T≤0,T≥0) in the preferred equivalence class, and for any compact generator H T H ∈ T H inTH \in \mathscr{T}H∈T, there exists an integer n > 0 n > 0 n > 0n>0n>0 (which may depend on H H HHH and on the t t ttt-structure), satisfying
(1) H H HHH belongs to T n T ≤ n T <= n\mathscr{T} \leq nT≤n and Hom ( H , T n ) = 0 Hom ⁡ ( H , T ≤ − n ) = 0 Hom(H,T <= -n)=0\operatorname{Hom}(H, \mathscr{T} \leq-n)=0Hom⁡(H,T≤−n)=0;
(2) Every object X T 0 X ∈ T ≤ 0 X inT^( <= 0)X \in \mathscr{T}^{\leq 0}X∈T≤0 admits an exact triangle E X D E → X → D E rarr X rarr DE \rightarrow X \rightarrow DE→X→D with E E ∈ E inE \inE∈ H ¯ n [ n , n ] ⟨ H ⟩ ¯ n [ − n , n ] bar((:H:))_(n)^([-n,n])\overline{\langle H\rangle}_{n}^{[-n, n]}⟨H⟩¯n[−n,n] and with D T 1 D ∈ T ≤ − 1 D inT <= -1D \in \mathscr{T} \leq-1D∈T≤−1.
Moreover, if H H HHH is a compact generator, ( T 0 , T 0 ) ( T ≤ 0 , T ≥ 0 ) (T <= 0,T >= 0)(\mathscr{T} \leq 0, \mathscr{T} \geq 0)(T≤0,T≥0) is a t t ttt-structure, and there exists an integer n > 0 n > 0 n > 0n>0n>0 satisfying (1) and (2) above, then the t t ttt-structure ( T 0 , T 0 ) ( T ≤ 0 , T ≥ 0 ) (T <= 0,T >= 0)(\mathscr{T} \leq 0, \mathscr{T} \geq 0)(T≤0,T≥0) must belong to the preferred equivalence class.
Remark 5.6. Strangely enough, the value of Proposition 5.5 can be that it allows us to find an explicit t-structure in the preferred equivalence class.
Consider the case where X X XXX is a quasicompact, separated scheme. By Bondal and Van den Bergh [6, THEOREM 3.1.1(II)], we know that the category D q c ( X ) D q c ( X ) D_(qc)(X)\mathbf{D}_{\mathbf{q c}}(X)Dqc(X) has a compact generator, but in Example 3.2 we mentioned that the existence proof is not overly constructive, it does not give us a handle on any explicit compact generator. Let G G GGG be some compact generator. From Alonso, Jeremías, and Souto [1, THEOREM A.1], we know that the subcategory G ¯ ( , 0 ] ⟨ G ⟩ ¯ ( − ∞ , 0 ] bar((:G:))^((-oo,0])\overline{\langle G\rangle}^{(-\infty, 0]}⟨G⟩¯(−∞,0] of Black Box 2.2(2) is equal to T G 0 T G ≤ 0 T_(G)^( <= 0)\mathscr{T}_{G}^{\leq 0}TG≤0 for a unique t-structure ( T G 0 , T G 0 ) T G ≤ 0 , T G ≥ 0 (T_(G)^( <= 0),T_(G)^( >= 0))\left(\mathscr{T}_{G}^{\leq 0}, \mathscr{T}_{G}^{\geq 0}\right)(TG≤0,TG≥0) in the preferred equivalence class. But this does not leave us a whole lot wiser-the compact generator G G GGG is not explicit, hence neither is the t t t\mathrm{t}t-structure.
However, the combination of [33, THEOREM 5.8] and [29, LEMMA 3.5] tells us that the category D q c ( X ) D q c ( X ) D_(qc)(X)\mathbf{D}_{\mathbf{q c}}(X)Dqc(X) is approximable, and it so happens that the t t t\mathrm{t}t-structure used in the proof, that is, the t t t\mathrm{t}t-structure for which a compact generator H H HHH and an integer n > 0 n > 0 n > 0n>0n>0 satisfying (1) and (2) of Proposition 5.5 are shown to exist, happens to be the standard t-structure. From Proposition 5.5, we now deduce that the standard t t ttt-structure is in the preferred equivalence class.

6. STRUCTURE THEOREMS IN APPROXIMABLE TRIANGULATED CATEGORIES

An approximable triangulated category T T T\mathscr{T}T must have a compact generator G G GGG, and Definition 5.4 constructed for us a preferred equivalence class of t t ttt-structures-namely all
those equivalent to ( T G 0 , T G 0 ) T G ≤ 0 , T G ≥ 0 (T_(G)^( <= 0),T_(G)^( >= 0))\left(\mathscr{T}_{G}^{\leq 0}, \mathscr{T}_{G}^{\geq 0}\right)(TG≤0,TG≥0). Recall that, for any t-structure ( T 0 , T 0 ) T ≤ 0 , T ≥ 0 (T <= 0,T^( >= 0))\left(\mathscr{T} \leq 0, \mathscr{T}^{\geq 0}\right)(T≤0,T≥0), it is customary to define
T = n = 1 T n , T + = n = 1 T n , T b = T T + T − = ⋃ n = 1 ∞   T ≤ n , T + = ⋃ n = 1 ∞   T ≥ − n , T b = T − ∩ T + T^(-)=uuu_(n=1)^(oo)T^( <= n),quadT^(+)=uuu_(n=1)^(oo)T^( >= -n),quadT^(b)=T^(-)nnT^(+)\mathscr{T}^{-}=\bigcup_{n=1}^{\infty} \mathscr{T}^{\leq n}, \quad \mathscr{T}^{+}=\bigcup_{n=1}^{\infty} \mathscr{T}^{\geq-n}, \quad \mathscr{T}^{b}=\mathscr{T}^{-} \cap \mathscr{T}^{+}T−=⋃n=1∞T≤n,T+=⋃n=1∞T≥−n,Tb=T−∩T+
It is an easy exercise to show, directly from Definition 5.1, that equivalent t-structures give rise to identical T , T + T − , T + T^(-),T^(+)\mathscr{T}^{-}, \mathscr{T}^{+}T−,T+, and T b T b T^(b)\mathscr{T}^{b}Tb. Therefore triangulated categories with a single compact generator, and in particular approximable triangulated categories, have preferred subcategories T , T + T − , T + T^(-),T^(+)\mathscr{T}^{-}, \mathscr{T}^{+}T−,T+, and T b T b T^(b)\mathscr{T}^{b}Tb, which are intrinsic-they are simply those corresponding to any t t t\mathrm{t}t-structure in the preferred equivalence class. In the remainder of this survey, we will assume that T , T + T − , T + T^(-),T^(+)\mathscr{T}^{-}, \mathscr{T}^{+}T−,T+, and T b T b T^(b)\mathscr{T}^{b}Tb always stand for the preferred ones.
In the heuristics of Discussion 2.1(2), we told the reader that a t-structure ( T 0 , T 0 ) T ≤ 0 , T ≥ 0 (T <= 0,T^( >= 0))\left(\mathscr{T} \leq 0, \mathscr{T}^{\geq 0}\right)(T≤0,T≥0) is to be viewed as a metric on T T T\mathscr{T}T. In Definition 6.1 below, the heuristic is that we construct a full subcategory T c T c − T_(c)^(-)\mathscr{T}_{c}^{-}Tc−to be the closure of T c T c T^(c)\mathscr{T}^{c}Tc with respect to any of the (equivalent) metrics that come from t-structures in the preferred equivalence class.
Definition 6.1. Let T T T\mathscr{T}T be an approximable triangulated category. The full subcategory T c T c − T_(c)^(-)\mathscr{T}_{c}^{-}Tc− is given by
Ob ( T c ) = { F T | For every integer n > 0 and for every t-structure ( T 0 , T 0 ) in the preferred equivalence class, there exists an exact triangle E F D in T with E T c and D T n } Ob ⁡ T c − = F ∈ T  For every integer  n > 0  and for every t-structure  ( T ≤ 0 , T ≥ 0 )  in the preferred equivalence class,   there exists an exact triangle  E → F → D  in  T  with  E ∈ T c  and  D ∈ T ≤ − n Ob(T_(c)^(-))={F inT|[" For every integer "n > 0" and for every t-structure "],[(T <= 0","T >= 0)" in the preferred equivalence class, "],[" there exists an exact triangle "E rarr F rarr D" in "T],[" with "E inT^(c)" and "D inT <= -n]}\operatorname{Ob}\left(\mathscr{T}_{c}^{-}\right)=\left\{F \in \mathscr{T} \left\lvert\, \begin{array}{c} \text { For every integer } n>0 \text { and for every t-structure } \\ (\mathscr{T} \leq 0, \mathscr{T} \geq 0) \text { in the preferred equivalence class, } \\ \text { there exists an exact triangle } E \rightarrow F \rightarrow D \text { in } \mathscr{T} \\ \text { with } E \in \mathscr{T}^{c} \text { and } D \in \mathscr{T} \leq-n \end{array}\right.\right\}Ob⁡(Tc−)={F∈T| For every integer n>0 and for every t-structure (T≤0,T≥0) in the preferred equivalence class,  there exists an exact triangle E→F→D in T with E∈Tc and D∈T≤−n}
The full subcategory T c b T c b T_(c)^(b)\mathscr{T}_{c}^{b}Tcb is defined to be T c b = T c T b T c b = T c − ∩ T b T_(c)^(b)=T_(c)^(-)nnT^(b)\mathscr{T}_{c}^{b}=\mathscr{T}_{c}^{-} \cap \mathscr{T}^{b}Tcb=Tc−∩Tb.
Remark 6.2. Let T T T\mathscr{T}T be an approximable triangulated category. Aside from the classical, full subcategory T c T c T^(c)\mathscr{T}^{c}Tc of compact objects, which we encountered back in Definition 1.1, we have in this section concocted five more intrinsic, full subcategories of T T T\mathscr{T}T : they are T T − T^(-)\mathscr{T}^{-}T−, T + , T b , T c T + , T b , T c − T^(+),T^(b),T_(c)^(-)\mathscr{T}^{+}, \mathscr{T}^{b}, \mathscr{T}_{c}^{-}T+,Tb,Tc−, and T c b T c b T_(c)^(b)\mathscr{T}_{c}^{b}Tcb. It can be proved that all six subcategories, that is, the old T c T c T^(c)\mathscr{T}^{c}Tc and the five new ones, are thick subcategories of T T T\mathscr{T}T. In particular, each of them is a triangulated category.
Example 6.3. It becomes interesting to figure out what all these categories come down to in examples.
Let X X XXX be a quasicompact, separated scheme. From Example 3.2, we know that the category T = D q c ( X ) T = D q c ( X ) T=D_(qc)(X)\mathscr{T}=\mathbf{D}_{\mathbf{q c}}(X)T=Dqc(X) is approximable, and in Remark 5.6 we noted that the standard t t t\mathrm{t}t-structure is in the preferred equivalence class. This can be used to show that, for T = D q c ( X ) T = D q c ( X ) T=D_(qc)(X)\mathscr{T}=\mathbf{D}_{\mathbf{q c}}(X)T=Dqc(X), we have
T = D q c ( X ) , T + = D q c + ( X ) , T b = D q c b ( X ) T c = D perf ( X ) , T c = D coh ( X ) , T c b = D coh b ( X ) , T − = D q c − ( X ) , T + = D q c + ( X ) , T b = D q c b ( X ) T c = D perf  ( X ) , T c − = D coh  − ( X ) , T c b = D coh  b ( X ) , {:[T^(-)=D_(qc)^(-)(X)","quadT^(+)=D_(qc)^(+)(X)","quadT^(b)=D_(qc)^(b)(X)],[T^(c)=D^("perf ")(X)","quadT_(c)^(-)=D_("coh ")^(-)(X)","quadT_(c)^(b)=D_("coh ")^(b)(X)","]:}\begin{aligned} & \mathscr{T}^{-}=\mathbf{D}_{\mathbf{q c}}^{-}(X), \quad \mathscr{T}^{+}=\mathbf{D}_{\mathbf{q c}}^{+}(X), \quad \mathscr{T}^{b}=\mathbf{D}_{\mathbf{q c}}^{b}(X) \\ & \mathscr{T}^{c}=\mathbf{D}^{\text {perf }}(X), \quad \mathscr{T}_{c}^{-}=\mathbf{D}_{\text {coh }}^{-}(X), \quad \mathscr{T}_{c}^{b}=\mathbf{D}_{\text {coh }}^{b}(X), \end{aligned}T−=Dqc−(X),T+=Dqc+(X),Tb=Dqcb(X)Tc=Dperf (X),Tc−=Dcoh −(X),Tcb=Dcoh b(X),
where the last two equalities assume that the scheme X X XXX is noetherian, and all six categories on the right of the equalities have their traditional meanings.
The reader can find an extensive discussion of the claims above in [31], more precisely in the paragraphs between [31, PROPOSITION 8.10] and [31, THEOREM 8.16]. That discussion
goes beyond the scope of the current survey, it analyzes the categories T c b T c T c b ⊂ T c − T_(c)^(b)subT_(c)^(-)\mathscr{T}_{c}^{b} \subset \mathscr{T}_{c}^{-}Tcb⊂Tc−in the generality of non-noetherian schemes, where they still have a classical description-of course, not involving the category of coherent sheaves. After all coherent sheaves do not behave well for non-noetherian schemes.
Remark 6.4. In this survey we spent some effort introducing the notion of approximable triangulated categories. In Example 3.2 we told the reader that it is a theorem (and not a trivial one) that, as long as a scheme X X XXX is quasicompact and separated, the derived category D q c ( X ) D q c ( X ) D_(qc)(X)\mathbf{D}_{\mathbf{q c}}(X)Dqc(X) is approximable. In this section we showed that every approximable triangulated category comes with canonically defined, intrinsic subcategories T , T + , T b , T c , T c T − , T + , T b , T c , T c − T^(-),T^(+),T^(b),T^(c),T_(c)^(-)\mathscr{T}^{-}, \mathscr{T}^{+}, \mathscr{T}^{b}, \mathscr{T}^{c}, \mathscr{T}_{c}^{-}T−,T+,Tb,Tc,Tc−, and T c b T c b T_(c)^(b)\mathscr{T}_{c}^{b}Tcb, and in Example 6.3 we informed the reader that, in the special case where T = D q c ( X ) T = D q c ( X ) T=D_(qc)(X)\mathscr{T}=\mathbf{D}_{\mathbf{q c}}(X)T=Dqc(X), these turn out to be D q c ( X ) , D q c + ( X ) , D q c b ( X ) , D perf ( X ) , D coh ( X ) D q c − ( X ) , D q c + ( X ) , D q c b ( X ) , D perf  ( X ) , D coh  − ( X ) D_(qc)^(-)(X),D_(qc)^(+)(X),D_(qc)^(b)(X),D^("perf ")(X),D_("coh ")^(-)(X)\mathbf{D}_{\mathbf{q c}}^{-}(X), \mathbf{D}_{\mathbf{q c}}^{+}(X), \mathbf{D}_{\mathbf{q c}}^{b}(X), \mathbf{D}^{\text {perf }}(X), \mathbf{D}_{\text {coh }}^{-}(X)Dqc−(X),Dqc+(X),Dqcb(X),Dperf (X),Dcoh −(X), and D coh b ( X ) D coh  b ( X ) D_("coh ")^(b)(X)\mathbf{D}_{\text {coh }}^{b}(X)Dcoh b(X), respectively.
Big deal. This teaches us that the traditional subcategories D q c ( X ) , D q c + ( X ) , D q c b ( X ) D q c − ( X ) , D q c + ( X ) , D q c b ( X ) D_(qc)^(-)(X),D_(qc)^(+)(X),D_(qc)^(b)(X)\mathbf{D}_{\mathbf{q c}}^{-}(X), \mathbf{D}_{\mathbf{q c}}^{+}(X), \mathbf{D}_{\mathbf{q c}}^{b}(X)Dqc−(X),Dqc+(X),Dqcb(X), D perf ( X ) , D coh ( X ) D perf  ( X ) , D coh  − ( X ) D^("perf ")(X),D_("coh ")^(-)(X)\mathbf{D}^{\text {perf }}(X), \mathbf{D}_{\text {coh }}^{-}(X)Dperf (X),Dcoh −(X), and D coh b ( X ) D coh  b ( X ) D_("coh ")^(b)(X)\mathbf{D}_{\text {coh }}^{b}(X)Dcoh b(X) of the category D q c ( X ) D q c ( X ) D_(qc)(X)\mathbf{D}_{\mathbf{q c}}(X)Dqc(X) all have intrinsic descriptions. This might pass as a curiosity, unless we can actually use it to prove something we care about that we did not use to know.
Discussion 6.5. To motivate the next theorem, it might help to think of the parallel with functional analysis.
Let M ( R ) M ( R ) M(R)M(\mathbb{R})M(R) be the vector space of Lebesgue-measurable, real-valued functions on R R R\mathbb{R}R. Given any two functions f , g M ( R ) f , g ∈ M ( R ) f,g in M(R)f, g \in M(\mathbb{R})f,g∈M(R), we can pair them by integrating the product, that is, we form the pairing
f , g = f g d μ ⟨ f , g ⟩ = ∫ f g d μ (:f,g:)=int fgd mu\langle f, g\rangle=\int f g d \mu⟨f,g⟩=∫fgdμ
where μ μ mu\muμ is Lebesgue measure. This gives us a map
M ( R ) × M ( R ) , R { } M ( R ) × M ( R ) → ⟨ − , − ⟩ R ∪ { ∞ } M(R)xx M(R)rarr"(:-,-:)"Ruu{oo}M(\mathbb{R}) \times M(\mathbb{R}) \xrightarrow{\langle-,-\rangle} \mathbb{R} \cup\{\infty\}M(R)×M(R)→⟨−,−⟩R∪{∞}
where the integral is declared to be infinite if it does not converge.
We can restrict this pairing to subspaces of M ( R ) M ( R ) M(R)M(\mathbb{R})M(R). For example, if f L p ( R ) f ∈ L p ( R ) f inL^(p)(R)f \in L^{p}(\mathbb{R})f∈Lp(R) and g L q ( R ) g ∈ L q ( R ) g inL^(q)(R)g \in L^{q}(\mathbb{R})g∈Lq(R) with 1 p + 1 q = 1 1 p + 1 q = 1 (1)/(p)+(1)/(q)=1\frac{1}{p}+\frac{1}{q}=11p+1q=1 then the integral converges, that is, f , g R ⟨ f , g ⟩ ∈ R (:f,g:)inR\langle f, g\rangle \in \mathbb{R}⟨f,g⟩∈R, and we deduce a map
L p ( R ) Hom ( L q ( R ) , R ) L p ( R ) ⟶ Hom ⁡ L q ( R ) , R L^(p)(R)longrightarrow Hom(L^(q)(R),R)L^{p}(\mathbb{R}) \longrightarrow \operatorname{Hom}\left(L^{q}(\mathbb{R}), \mathbb{R}\right)Lp(R)⟶Hom⁡(Lq(R),R)
which turns out to be an isometry of Banach spaces.
The category-theoretic version is that on any category T T T\mathscr{T}T there is the pairing sending two objects A , B T A , B ∈ T A,B inTA, B \in \mathscr{T}A,B∈T to Hom ( A , B ) Hom ⁡ ( A , B ) Hom(A,B)\operatorname{Hom}(A, B)Hom⁡(A,B). Of course, this pairing is not symmetric, we have to keep track of the position of A A AAA and of B B BBB in Hom ( A , B ) Hom ⁡ ( A , B ) Hom(A,B)\operatorname{Hom}(A, B)Hom⁡(A,B). If R R RRR is a commutative ring and T T T\mathscr{T}T happens to be an R R RRR-linear category, then Hom ( A , B ) Hom ⁡ ( A , B ) Hom(A,B)\operatorname{Hom}(A, B)Hom⁡(A,B) is an R R RRR-module and the pairing delivers a map
T o p × T Hom ( , ) R Mod T o p × T → Hom ⁡ ( − , − ) R − Mod T^(op)xxTrarr"Hom(-,-)"R-Mod\mathscr{T}^{\mathrm{op}} \times \mathscr{T} \xrightarrow{\operatorname{Hom}(-,-)} R-\operatorname{Mod}Top×T→Hom⁡(−,−)R−Mod
where the op keeps track of the variable in the first position. And now we are free to restrict to subcategories of T T T\mathscr{T}T.
If T T T\mathscr{T}T happens to be approximable and R R RRR-linear, we have just learned that it comes with six intrinsic subcategories T , T + , T b , T c , T c T − , T + , T b , T c , T c − T^(-),T^(+),T^(b),T^(c),T_(c)^(-)\mathscr{T}^{-}, \mathscr{T}^{+}, \mathscr{T}^{b}, \mathscr{T}^{c}, \mathscr{T}_{c}^{-}T−,T+,Tb,Tc,Tc−, and T c b T c b T_(c)^(b)\mathscr{T}_{c}^{b}Tcb. We are free to restrict the Hom pairing to any couple of them. This gives us 36 possible pairings, and each of those yields two maps from a subcategory to the dual of another. There are 72 cases we could study, and the theorem below tells us something useful about four of those.
Theorem 6.6. Let R R RRR be a noetherian ring, and let T T T\mathscr{T}T be an R R RRR-linear, approximable triangulated category. Suppose there exists in T T T\mathscr{T}T a compact generator G G GGG so that Hom ( G , G [ n ] ) Hom ⁡ ( G , G [ n ] ) Hom(G,G[n])\operatorname{Hom}(G, G[n])Hom⁡(G,G[n]) is a finite R R RRR-module for all n Z n ∈ Z n inZn \in \mathbb{Z}n∈Z. Consider the two functors
Y : T c Hom R ( [ T c ] o p , R M o d ) , Y ~ : [ T c ] o p Hom R ( T c b , R M o d ) Y : T c − → Hom R ⁡ T c o p , R − M o d , Y ~ : T c − o p → Hom R ⁡ T c b , R − M o d Y:T_(c)^(-)rarrHom_(R)([T^(c)]^(op),R-Mod),quad widetilde(Y):[T_(c)^(-)]^(op)rarrHom_(R)(T_(c)^(b),R-Mod)\mathscr{Y}: \mathscr{T}_{c}^{-} \rightarrow \operatorname{Hom}_{R}\left(\left[\mathscr{T}^{c}\right]^{\mathrm{op}}, R-\mathrm{Mod}\right), \quad \widetilde{\mathscr{Y}}:\left[\mathscr{T}_{c}^{-}\right]^{\mathrm{op}} \rightarrow \operatorname{Hom}_{R}\left(\mathscr{T}_{c}^{b}, R-\mathrm{Mod}\right)Y:Tc−→HomR⁡([Tc]op,R−Mod),Y~:[Tc−]op→HomR⁡(Tcb,R−Mod)
defined by the formulas Y ( B ) = Hom ( , B ) Y ( B ) = Hom ⁡ ( − , B ) Y(B)=Hom(-,B)\mathscr{Y}(B)=\operatorname{Hom}(-, B)Y(B)=Hom⁡(−,B) and Y ~ ( A ) = Hom ( A , ) Y ~ ( A ) = Hom ⁡ ( A , − ) widetilde(Y)(A)=Hom(A,-)\widetilde{\mathscr{Y}}(A)=\operatorname{Hom}(A,-)Y~(A)=Hom⁡(A,−), as in Discussion 6.5 . Now consider the following composites:
T c b c T c Y Hom R ( [ T c ] o p , R M o d ) , [ T c ] o p = i ~ [ T c ] o p Y ~ Hom R ( T c b , R Mod ) . T c b c T c − → Y Hom R ⁡ T c o p , R − M o d , T c o p = i ~ T c − o p → Y ~ ⟶ Hom R ⁡ T c b , R − Mod . {:[T_(c)^(bc)T_(c)^(-)rarr"Y"Hom_(R)([T^(c)]^(op),R-Mod)","],[{:[T^(c)]^(op)=_(" tilde(i)")^("")[T_(c)^(-)]^(op)rarr_(" widetilde(Y)")^("longrightarrow")Hom_(R)(T_(c)^(b),R-Mod).:}]:}\begin{gathered} \mathscr{T}_{c}^{b c} \mathscr{T}_{c}^{-} \xrightarrow{\mathscr{Y}} \operatorname{Hom}_{R}\left(\left[\mathscr{T}^{c}\right]^{\mathrm{op}}, R-\mathrm{Mod}\right), \\ {\left[\mathscr{T}^{c}\right]^{\mathrm{op}} \xlongequal[\tilde{i}]{ }\left[\mathscr{T}_{c}^{-}\right]^{\mathrm{op}} \xrightarrow[\widetilde{\mathscr{Y}}]{\longrightarrow} \operatorname{Hom}_{R}\left(\mathscr{T}_{c}^{b}, R-\operatorname{Mod}\right) .} \end{gathered}TcbcTc−→YHomR⁡([Tc]op,R−Mod),[Tc]op=i~[Tc−]op→Y~⟶HomR⁡(Tcb,R−Mod).
We assert:
(1) The functor Y Y Y\mathscr{Y}Y is full, and the essential image consists of the locally finite homological functors (see Explanation 6.7 for the definition of locally finite functors). The composite Y Y Y\mathscr{Y}Y â—‹ i i iii is fully faithful, and the essential image consists of the finite homological functors (again, see Explanation 6.7 for the definition).
(2) With the notation as in Black Box 2.2(1), assume 1 1 ^(1){ }^{1}1 that T = H ¯ T = ⟨ H ⟩ ¯ T= bar((:H:))\mathscr{T}=\overline{\langle H\rangle}T=⟨H⟩¯ for some integer n > 0 n > 0 n > 0n>0n>0 and some object H T c b H ∈ T c b H inT_(c)^(b)H \in \mathscr{T}_{c}^{b}H∈Tcb. Then the functor Y ~ Y ~ widetilde(Y)\widetilde{\mathscr{Y}}Y~ is full, and the essential image consists of the locally finite homological functors. The composite Y ~ Y ~ widetilde(Y)\widetilde{\mathscr{Y}}Y~ â—‹ г̃ is fully faithful, and the essential image consists of the finite homological functors.
Explanation 6.7. In the statement of Theorem 6.6, the locally finite functors, either of the form H : [ T c ] op R H : T c op  → R H:[T^(c)]^("op ")rarr RH:\left[\mathscr{T}^{c}\right]^{\text {op }} \rightarrow RH:[Tc]op →R-Mod or of the form H : T c b R H : T c b → R H:T_(c)^(b)rarr RH: \mathscr{T}_{c}^{b} \rightarrow RH:Tcb→R-Mod, are the functors such that
(1) H ( A [ i ] ) H ( A [ i ] ) H(A[i])H(A[i])H(A[i]) is a finite R R RRR-module for every i Z i ∈ Z i inZi \in \mathbb{Z}i∈Z and every A A AAA in either T c T c T^(c)\mathscr{T}^{c}Tc or T c b T c b T_(c)^(b)\mathscr{T}_{c}^{b}